15. Q15b
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Similar Question 1
<p>A parabola has equation <code class='latex inline'>\displaystyle y = 2(x + 6)^2 -2 </code></p><p>a) Expand and simplify to write the equation in the form <code class='latex inline'>y =ax^2 + bx + c</code>.</p><p>b) Factor your equation from part a).</p><p>c) Do the three equations represent the same parabola? Justify your response.</p>
Similar Question 2
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, opens upward, and has vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Now find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>translated 2 units to the left</p>
Similar Question 3
<p>A graphing calculator was used together with the vertex form <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> to graph the screens shown. For the set of graphs on each screen, tell which of the variables <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code> remained constant and which changed. Give possible values for the variables that remained constant.</p><p>M</p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>A parabola has equation <code class='latex inline'>y=(x+4)^2</code>.</p><p>a) Identify the coordinates of the vertex.</p><p>b) Expand and simplify the equation.</p><p>c) Verify that the coordinates of the vertex satisfy the equation from part b).</p>
<p>Describe how the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of the given quadratic functions differ from the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of corresponding points of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p><p><code class='latex inline'>\displaystyle y=-\frac{1}{2} x^{2}-4 </code></p>
<p>Determine whether each quadratic function intersects the <code class='latex inline'>x</code>—axis at one point, two points, or not at all. Do not draw the graph.</p><p><code class='latex inline'>g(x)=-3(x-5)^2+2</code></p>
<p>Determine any x-intercepts, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle y=x^{2}+6 </code></p>
<p>Determine any x-intercepts, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle y=0.5 x^{2}-3 </code></p>
<p>Write and equation for the parabola with vertex (-4,5), opening downward, and with a vertical stretch of factor 3. </p>
<p>Determine the maximum or minimum value. Use at least two different methods.</p><p><code class='latex inline'> \displaystyle g(x) = -2(x + 1)^2 - 5 </code></p>
<p>Describe the graph of <code class='latex inline'> y=a(x-h)^{2}+k </code> if</p><p> <code class='latex inline'> h=0 </code> </p>
<p>For the function, state the vertex and whether the function has a maximum or minimum value. Explain how you decided.</p><p><code class='latex inline'> \displaystyle f(x) = (x + 5)^2-3 </code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex at <code class='latex inline'>\displaystyle (0,0) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>compressed vertically by 3 , reflected in the <code class='latex inline'>\displaystyle x </code>-axis, and translated 2 units up</p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, opens upward, and has vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Now find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p> stretched vertically by a factor of 5</p>
<p>Write each function in standard form.</p><p><code class='latex inline'>\displaystyle f(x)=4(x-5)^{2}+9 </code></p>
<p>Write the equation of a parabola that matches each description.</p><p>The graph of <code class='latex inline'>y=x^2</code> is compressed vertically by a factor of and then translated 6 units down. 5</p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = (x+2)^2 + 5</code></p>
<p>Create a table of values for each quadratic relation, and sketch its graph. Then determine</p> <ul> <li>i) the equation of the axis of symmetry</li> <li>ii) the coordinates of the vertex</li> <li>iii) the y-intercept</li> <li>iv) the zeros</li> <li>v) the maximum or minimum value</li> </ul> <p><code class='latex inline'>y=-x^2-1</code></p>
<p>Write an equation of a parabola that satisfies each set of conditions. </p><p>opens downward, congruent with <code class='latex inline'>\displaystyle y=\frac{1}{3} x^{2} </code>, vertex <code class='latex inline'>\displaystyle (-2,3) </code></p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle f(x)=5 x^{2} </code></p>
<p>a) Graph the function <code class='latex inline'>f(x) = (x + 3)^{2} - 2</code>.<br> b) Graph <code class='latex inline'>g(x)</code>, which is the reflection of <code class='latex inline'>f(x)</code> in the <code class='latex inline'>y</code>-axis. Write the equation of <code class='latex inline'>g(x)</code>.<br> c) Determine a translation that can be applied to <code class='latex inline'>f(x)</code> to obtain <code class='latex inline'>g(x)</code>.<br> d) Verify algebraically that the transformations in parts b) and c) are the same.<br> e) Predict if the same would be true for reflections in the <code class='latex inline'>x</code>-axis. Explain.<br> f) Would this work for any other type of function? Explain. </p>
<p>Determine any x-intercepts, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle y=x^{2}-2 </code></p>
<p>A parabola has equation <code class='latex inline'>y=(x-3)^2</code>.</p><p>a) Identify the coordinates of the vertex.</p><p>b) Expand and simplify the equation.</p><p>c) Verify that the coordinates of the vertex satisfy the equation from part b).</p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle y=-\frac{4}{9} x^{2} </code></p>
<p>Sketch a graph of each parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.</p><p><code class='latex inline'>\displaystyle y = 2(x-5)^2 </code></p>
<p>How are the parabolas <code class='latex inline'>f(x) = - 3(x - 2)^2 - 4</code> and <code class='latex inline'>g(x) = 6(x - 2)^2 - 4</code> the same? How are they different? </p>
<p>The following transformations are applied to a parabola with the equation <code class='latex inline'>y=2(x+3)^2-1</code> Determine the equation that will result after each transformation.</p> <ul> <li>a reflection in the <code class='latex inline'>x</code>-axis</li> </ul>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph <code class='latex inline'>y=x^2</code> is stretch vertically by a factor of 3. </p>
<p>Match each graph with the correct equation. The graph of <code class='latex inline'>\displaystyle y=x^{2} </code> is in green in each diagram.</p><p>a) <code class='latex inline'>\displaystyle y=4 x^{2} </code></p><p>c) <code class='latex inline'>\displaystyle y=\frac{2}{3} x^{2} </code></p><p>b) <code class='latex inline'>\displaystyle y=-3 x^{2} </code></p><p>d) <code class='latex inline'>\displaystyle y=-0.4 x^{2} </code></p><img src="/qimages/145217" />
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = -(x+8)^2 - 4</code></p>
<p>Write the equation of the parabola in vertex form.</p><img src="/qimages/1545" />
<p>A parabola has equation <code class='latex inline'>\displaystyle y = 2(x + 6)^2 -2 </code></p><p>a) Expand and simplify to write the equation in the form <code class='latex inline'>y =ax^2 + bx + c</code>.</p><p>b) Factor your equation from part a).</p><p>c) Do the three equations represent the same parabola? Justify your response.</p>
<p>Sketch the graph of the parabola. Describe the transformation from the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'> \displaystyle y = -0.5x^2 </code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, opens upward, and has vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Now find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>compressed by a factor of <code class='latex inline'>\displaystyle \frac{1}{2} </code></p>
<p>Match each translation with the correct quadratic relation.</p><p>a) 3 units left, 4 units down</p><p>b) 2 units right, 4 units down</p><p>c) 5 units left</p><p>d) 3 units right, 2 units up</p> <ul> <li>i) <code class='latex inline'>y = (x - 3)^2 + 2</code></li> <li>ii) <code class='latex inline'>y = (x + 3)^2 - 4</code></li> <li>iii) <code class='latex inline'>y = (x - 2)^2 - 4</code></li> <li>iv) <code class='latex inline'>y = (x + 5)^2 </code></li> </ul>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is translated 2 units to the left and translated 3 units up</p>
<p>Write in standard form. </p><p><code class='latex inline'> \displaystyle f(x) = (x + 3)^2 -7 </code></p>
<p> Write an equation of the quadratic equation that satisfies each set of conditions.</p><p>The parabola is congruent to <code class='latex inline'>y = 2x^2</code>, and has a maximum value of 14.</p>
<p>Write one sentence that compares each pair of graphs. </p><p><code class='latex inline'>\displaystyle y=2 x^{2}+7 </code> and <code class='latex inline'>\displaystyle y=2 x^{2}-2 </code></p>
<p> Find the missing value.</p><p>The parabola <code class='latex inline'>y = 3(x - 2)^2 + k</code> contains the point <code class='latex inline'>(-2, 1)</code>.</p>
<p>Describe how the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of the given quadratic functions differ from the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of corresponding points of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p><p><code class='latex inline'>\displaystyle y=-2(x-4)^{2} </code></p>
<p>Describe the graph of <code class='latex inline'> y=a(x-h)^{2}+k </code> if</p><p> <code class='latex inline'> h=0 </code> and <code class='latex inline'> k=0 </code> </p>
<p>Use transformations to determine the vertex, axis of symmetry, and direction of opening of each parabola. Sketch the graph.</p><p> <code class='latex inline'>\displaystyle y=x^{2}-7 </code></p>
<p>If <code class='latex inline'>\displaystyle y=x^{2} </code> is the base curve for the graphs shown, what equations could be used to produce these screens on a graphing calculator? The scale on both axes is 1 unit per tick mark.</p><img src="/qimages/163471" />
<p>Identify the transformation(s) that must be applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to create a graph of each equation. Then state the coordinates of the image of the point <code class='latex inline'>\displaystyle (2,4) </code>.</p><p><code class='latex inline'>\displaystyle y=\frac{1}{5} x^{2} </code></p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = (x-4)^2</code></p>
<p>Match each equation to the corresponding graph.</p><p>a) <code class='latex inline'>\displaystyle f(x)=(x-4)^{2} </code></p><p>b) <code class='latex inline'>\displaystyle g(x)=(x-6)^{2}-2 </code></p><p>c) <code class='latex inline'>\displaystyle h(x)=(x+6)^{2}+3 </code></p><p>d) <code class='latex inline'>\displaystyle k(x)=x^{2}-5 </code></p><img src="/qimages/155191" />
<p>Sketch a graph of each parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.</p><p><code class='latex inline'>\displaystyle y = x^2 - 6 </code></p>
<p>Find an equation for the parabola with vertex (6,4) that passes through the point (8,2). </p>
<p>Write an equation for the parabola with vertex (3,5), opening upward, and with no vertical stretch or compression. </p>
<p>Determine the equation of the parabola with vertex <code class='latex inline'>(-2, 5)</code> and that passes through <code class='latex inline'>(4, -8)</code></p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = (x+3)^2</code></p>
<p>The points <code class='latex inline'>(-2, -12)</code> and (2, 4) lie on the parabola <code class='latex inline'>y = a(x - 1)^2 + k</code>. What is the vertex of this parabola?</p>
<p>Find </p> <ul> <li>Vertex</li> <li>Axis of Symmetry</li> <li>Determine if it opens up/down</li> <li>Range</li> <li>Sketch</li> </ul> <p><code class='latex inline'> \displaystyle f(x) = -2(x - 4)^2 + 1 </code></p>
<p>Write the equations of two different quadratic relations that match each description.</p><p>The graph has a wider opening than the graph of <code class='latex inline'>y =-0.5x^2</code></p>
<p>Write the equation of the quadratic function, first in vertex form and then in standard form.</p><p>vertex (3, 5) and passing through (1, 1)</p>
<p>Write in standard form. </p><p><code class='latex inline'> \displaystyle f(x) = 2(x -1)^2 + 5 </code></p>
<p>Describe the transformation(s) that were applied to the graph of <code class='latex inline'>y = x^2</code> to obtain the graph not labelled <code class='latex inline'>y = x^2</code>. Write the equation of the black graph.</p><img src="/qimages/3043" />
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is translated 3 units to the right and translate 1 unit down</p>
<p>Without solving, determine the number of real roots it has.</p><p><code class='latex inline'>y = (x-3)^2</code></p>
<p>Write the equation of a parabola with each set of properties.</p> <ul> <li>axis of symmetry <code class='latex inline'>x = 4</code>, opens upward, two zeros, narrower than <code class='latex inline'>y=x^2</code></li> </ul>
<p>If <code class='latex inline'>\displaystyle y=x^{2}-4 </code>, what value(s) of <code class='latex inline'>\displaystyle x </code> give each of the following values of <code class='latex inline'>\displaystyle y ? </code></p><p><code class='latex inline'>-4</code></p>
<p>Write one sentence that compares each pair of graphs. </p><p><code class='latex inline'>\displaystyle y=-x^{2} </code> and <code class='latex inline'>\displaystyle y=-x^{2}+5 </code></p>
<p>Determine the zeros of each function.</p><p><code class='latex inline'> \displaystyle g(x) = 3(x - 1)^2 </code></p>
<p>Error Analysis Your friend wrote the transformations shown to describe how to change the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} </code> to the graph of <code class='latex inline'>\displaystyle g(x)=2(x+1)^{2}-3 . </code> Explain the error and give the correct transformations.</p><img src="/qimages/89540" />
<p>Sketch each parabola.</p><p><code class='latex inline'>\displaystyle y=(2+x)^{2} </code></p>
<p>Write an equation for each parabola. </p><img src="/qimages/22258" />
<p>For the function, state the vertex and whether the function has a maximum or minimum value. Explain how you decided.</p><p><code class='latex inline'> \displaystyle f(x) =-(x + 1)^2 + 6 </code></p>
<p>For each function, identify the horizontal translation of the parent function, <code class='latex inline'>\displaystyle f(x)=x^{2} </code>. Then graph the function.</p><p><code class='latex inline'>\displaystyle y=(x+3)^{2} </code></p>
<p>Write the equation of the parabola in vertex form.</p><img src="/qimages/1541" />
<p>Graph </p><p><code class='latex inline'>\displaystyle f(x) = -3(x - 2)^2 + 5 </code></p>
<p>Determine the vertex, the axis of symmetry, the direction the parabola opens, and the number of zeros for each quadratic function. Sketch a graph of each.</p><p><code class='latex inline'> \displaystyle f(x) =-2(x +3)^2 -1 </code></p>
<p>Find <code class='latex inline'> a </code> and <code class='latex inline'> k </code> so that the given points lie on the parabola.</p><p><code class='latex inline'>\displaystyle y=a(x-4)^{2}+k ;(1,-13),(-1,-45) </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions.</p><p>opens upward, narrower than <code class='latex inline'>\displaystyle y=x^{2} </code>, and vertex <code class='latex inline'>\displaystyle (2,0) </code></p>
<p>You investigated the graphs of <code class='latex inline'>y = (x - h)^2</code>. Consider the quadratic relation <code class='latex inline'>y = (x - 5)^2</code>.</p><p>Write the coordinates of the vertex of the parabola.</p>
<p>Which equation represents the graph shown? Explain your reasoning.</p> <ul> <li>a) <code class='latex inline'>y = -3(x + 3)^2 + 8</code></li> <li>b) <code class='latex inline'>y = -3(x - 3)^2 + 8</code></li> <li>c) <code class='latex inline'>y = 3(x - 3)^2 -8</code></li> <li>d) <code class='latex inline'>y = -2(x - 3)^2 +8</code></li> </ul> <img src="/qimages/6233" />
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y = x^2</code> is translated 3 units downward. </p>
<p>Write an equation for a parabola with the given vertex and given value of <code class='latex inline'> a </code> .</p><p><code class='latex inline'>\displaystyle (0,-7) ; a=-8 </code></p>
<p>Use a graphing calculator or graphing software to graph each function. State its domain and range. <code class='latex inline'>\displaystyle h(x)=2(x-1)^{2}+3 </code></p>
<p>Describe the transformations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain a graph of each quadratic relation. Sketch the graph by hand. Start with the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> and use the appropriate transformations.</p><p><code class='latex inline'>\displaystyle y=2(x+1)^{2}+4 </code></p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle f(x)=2 \frac{1}{4} x^{2} </code></p>
<p>Write the equation of a parabola with each set of properties.</p> <ul> <li>vertex at <code class='latex inline'>(-3,5)</code>, opens downward, wider than <code class='latex inline'>y=x^2</code></li> </ul>
<p>The following transformations are applied to the graph of <code class='latex inline'>y=x^2</code> Determine the equation of each new relation.</p><p><strong>a)</strong> vertical stretch by a factor of 4</p><p><strong>b)</strong> a translation of 3 units left</p><p><strong>c)</strong> a reflection in the x—axis, followed by a translation 2 units up</p><p><strong>d)</strong> a vertical compression by a factor of <code class='latex inline'>\displaystyle{\frac{1}{2}}</code></p><p><strong>e)</strong> a translation of 5 units right and 4 units down</p><p><strong>f)</strong> a vertical stretch by a factor of 2, followed by a reflection in the x—axis and a translation 1 unit left</p>
<p>Sketch the graph of each equation by correctly applying the required transformation(s) to points on the graph of <code class='latex inline'>y = x^2</code>. Use a separate grid for each graph.</p><p><code class='latex inline'> \displaystyle y = \frac{2}{3}x^2 </code></p>
<p>For each function, determine the vertex and two points that satisfy the equation. Use the information to sketch the graph of each.</p><p><code class='latex inline'> \displaystyle y = (x - 4)^2 </code></p>
<p>The vertex is <code class='latex inline'>(4, 0)</code>, and the y-intercept is <code class='latex inline'>8</code>.</p>
<p>Describe how the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of the given quadratic functions differ from the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of corresponding points of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p><p><code class='latex inline'>\displaystyle y=x^{2}+7 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions. </p><p>vertex <code class='latex inline'>\displaystyle (2,-2), x </code>-intercepts of 0 and 4</p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle 4.3 x^{2}+y=-0.5 </code></p>
<p>Write the equation of a parabola that matches each description.</p><p>The graph of <code class='latex inline'>y=x^2</code> is reflected about the <code class='latex inline'>x-</code>axis, stretched vertically by a factor of 6, translated 3 units right, and translated 4 units up.</p>
<p>A parabola has vertex (-3, 7), and one <code class='latex inline'>x</code>-intercept is <code class='latex inline'>-11</code>. Find the other <code class='latex inline'>x</code>-intercept and the <code class='latex inline'>y</code>—intercept.</p>
<p>The vertex of a quadratic relation is (4,-12).</p> <ul> <li>Write an equation to describe all parabolas with this vertex.</li> </ul>
<p>For the function, state the vertex and whether the function has a maximum or minimum value. Explain how you decided.</p><p><code class='latex inline'> \displaystyle f(x) =2(x - 3)^2 - 5 </code></p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y-9.9=1.6 x^{2} </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions.</p><p> opens upward, congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, and vertex <code class='latex inline'>\displaystyle (0,4) </code></p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle y=\frac{2}{5} x^{2} </code></p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=3 x^{2} </code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, opens upward, and has vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Now find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>reflected in the <code class='latex inline'>\displaystyle x </code>-axis and translated 2 units to the right and 4 units down</p>
<p>Describe how each transformation or sequence fo transformations of the function <code class='latex inline'>f(x) = 3x^2</code> will affect the number of zeros of the function has.</p><p>a) a vertical stretch of factor 2.</p><p>b) a horizontal translation 3 units to the left.</p><p>c) a horizontal compression of factor 2 and the na reflection in the x-axis.</p><p>d) a vertical translation 3 units down.</p><p>e) a horizontal translation 4 units to the right and then a vertical translation 3 units up.</p><p>f) a reflection in the x-axis, then a horizontal translation 1 unit to the left, and then a vertical translation 5 units up.</p>
<p>For each function <code class='latex inline'>g(x)</code>, identify the base function as one of <code class='latex inline'>f(x) = x, f(x) = x^{2}, f(x) = \sqrt{x},</code> or <code class='latex inline'>f(x) = \frac{1}{x}</code>, and describe the transformation in the form <code class='latex inline'>y = f(x - d) + c</code> and in words. Then sketch a graph of <code class='latex inline'>g(x)</code> and state the domain and range of each function. </p><p><code class='latex inline'>g(x) = (x - 5)^{2}</code></p>
<p>Determine whether each quadratic function intersects the <code class='latex inline'>x</code>—axis at one point, two points, or not at all. Do not draw the graph.</p><p><code class='latex inline'>g(x)=-3(x+2)^2</code></p>
<ol> <li>The vertex of a parabola is <code class='latex inline'>\displaystyle (-2,-4) . </code> One <code class='latex inline'>\displaystyle x </code>-intercept is <code class='latex inline'>\displaystyle 7 . </code> What is the other <code class='latex inline'>\displaystyle x </code>-intercept?</li> </ol>
<p>Identify the transformation(s) that must be applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to create a graph of each equation. Then state the coordinates of the image of the point <code class='latex inline'>\displaystyle (2,4) </code>.</p><p><code class='latex inline'>\displaystyle y=4 x^{2} </code></p>
<p>Determine an equation to represent the parabola.</p><img src="/qimages/8180" />
<p>Find an equation for the parabola with vertex (2,6) that passes through the point (5,3). </p>
<p>Write in standard form. </p><p><code class='latex inline'> \displaystyle f(x) = -(x + 7)^2 +3 </code></p>
<p>Write an equation for each parabola. </p><img src="/qimages/22263" />
<ol> <li>Find the value of <code class='latex inline'>\displaystyle k </code> so that the parabola <code class='latex inline'>\displaystyle y=-2 x^{2}+k </code> passes through the point <code class='latex inline'>\displaystyle (-3,-33) </code>.</li> </ol>
<p>What is the vertex of <code class='latex inline'>\displaystyle y=(x+1)^{2}-4 </code>, and what is the equation of the axis of symmetry? <code class='latex inline'>\displaystyle \begin{array}{ll}\text { A. }(1,-4) ; x=1 & \text { C. }(1,4) ; x=1 \\ \text { B. }(-1,-4) ; x=-1 & \text { D. }(-1,4) ; x=-1\end{array} </code></p>
<p>Identify the transformation(s) that must be applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to create a graph of each equation. Then state the coordinates of the image of the point <code class='latex inline'>\displaystyle (2,4) </code>.</p><p><code class='latex inline'>\displaystyle y=-\frac{2}{3} x^{2} </code></p>
<p>Describe the transformations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain the graph of each quadratic relation.</p><p><code class='latex inline'>\displaystyle y=2 x^{2}+7 </code></p>
<p>Describe the sequence of transformations needed to transform the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> into the graph of <code class='latex inline'>\displaystyle y=2 x^{2}-4 </code></p>
<p>Graph each function. The domain is the set of real numbers. Find the range.</p><p><code class='latex inline'>\displaystyle y=x^{2}-2 </code></p>
<p>Sketch the graphs below.</p><p><code class='latex inline'> \displaystyle y = 4(x -2)^2 - 5 </code></p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-1.5 x^{2} </code></p>
<p>Write the equation of the quadratic function, first in vertex form and then in standard form.</p><p>Vertex (— 6, — 5) and passing through (— 3, 4)</p>
<p>The following transformations are applied to a parabola with the equation <code class='latex inline'>y=2(x+3)^2-1</code> Determine the equation that will result after each transformation.</p> <ul> <li>a stretch by a factor of 6</li> </ul>
<ol> <li>Write an equation for the parabola with the given vertex and the given value of <code class='latex inline'>\displaystyle a </code>.</li> </ol> <p><code class='latex inline'>\displaystyle (7,0) ; a=1 </code></p>
<p>If <code class='latex inline'>\displaystyle y=x^{2}-4 </code>, what value(s) of <code class='latex inline'>\displaystyle x </code> give each of the following values of <code class='latex inline'>\displaystyle y ? </code></p><p><code class='latex inline'>0</code></p>
<p>Write the equation of the parabola in vertex form.</p><img src="/qimages/1546" />
<p>a) If <code class='latex inline'>\displaystyle y=x^{2} </code> is the base curve, write the equations of the parabolas that produce the following pattern shown on the calculator screen below. The scale on both axes is 1 unit per tick mark.</p><img src="/qimages/163435" /><p>b) Create your own pattern using parabolas, and write the associated equations. Use <code class='latex inline'>\displaystyle y=x^{2} </code> as the base parabola.</p>
<p>Graph each function, and complete the table.</p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|l|l|l|l|l|} \hline Factored Form & Standard Form & Axis of Symmetry & Zeros & y -intercept & Vertex & Maximum or Minimum Value \\ \hline & j(x)=4 x^{2}-121 & & & & \\ \hline \end{array} </code></p>
<p>Graph each function.</p><p><code class='latex inline'>\displaystyle y=-4 x^{2}+1 </code></p>
<p>For each function, identify the horizontal translation of the parent function, <code class='latex inline'>\displaystyle f(x)=x^{2} </code>. Then graph the function.</p><p><code class='latex inline'>\displaystyle y=(x-4)^{2} </code></p>
<p>For each quadratic relation, state the coordinates of the vertex and the direction of opening. Then, determine how many <code class='latex inline'>x</code>-intercepts the relation has.</p><p><code class='latex inline'>y=(x-5)^2</code></p>
<p>For each function, identify the horizontal translation of the parent function, <code class='latex inline'>\displaystyle f(x)=x^{2} </code>. Then graph the function.</p><p><code class='latex inline'>\displaystyle y=(x+1)^{2} </code></p>
<p>Describe the transformation(s) that were applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain each black graph. Write the equation of the black graph.</p><img src="/qimages/145221" />
<p>Examine each parabola.</p> <ul> <li>i) Determine the coordinates of the vertex.</li> <li>ii) Determine the zeros.</li> <li>iii) Determine the equation of the axis of symmetry.</li> <li>iv) If you calculated the second differences, would they be positive or negative? Explain.</li> </ul> <img src="/qimages/757" />
<p>The parabola <code class='latex inline'>y = x^2</code> is compressed vertically and translated down and right. The point <code class='latex inline'>(4, -10)</code> is on the new graph. What is a possible equation for the new graph?</p>
<p>Graph each parabola. State the coordinates of the vertex. Find any intercepts.</p><p><code class='latex inline'>\displaystyle y=x^{2}+1 </code></p>
<p>For each of the following, state the equation of a parabola congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> with the given property.</p><p>The graph is 4 units to the left and 5 units down from the graph of <code class='latex inline'>\displaystyle y=x^{2} </code></p>
<p>Determine any x-intercepts, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle y=2 x^{2}-10 </code></p>
<p>We rotated the parabola <code class='latex inline'>y = x^2</code> by 180° around a point. The new vertex is <code class='latex inline'>(6, -8)</code>. What is the equation of the new parabola?</p>
<p>State the vertex of each parabola and indicate the maximum or minimum value of the function.</p><img src="/qimages/883" />
<p>If <code class='latex inline'>\displaystyle y=x^{2} </code> is the base curve for the graphs shown, what equations could be used to produce these screens on a graphing calculator? The scale on both axes is 1 unit per tick mark.</p><img src="/qimages/163472" />
<p>Write each function in standard form.</p><p><code class='latex inline'>\displaystyle g(x)=-0.5(x-4)^{2}+2 </code></p>
<p>Liz describes the graph of <code class='latex inline'>\displaystyle y=-2(x-3)^{2}+4 </code> using transformations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} . </code> Which transformation is not correct?</p><p>A. vertical stretch by a factor of 2</p><p>B. reflection in the <code class='latex inline'>\displaystyle x </code>-axis</p><p>C. horizontal shift 3 units left</p><p>D. vertical shift 4 units up</p>
<p>Determine the answers to the following questions for each of the given transformed quadratic functions.</p><p>i) How does the shape of the graph compare with the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} ? </code></p><p>ii) What are the coordinates of the vertex and the equation of the axis of symmetry?</p><p>iii) Graph the transformed function and <code class='latex inline'>\displaystyle f(x)=x^{2} </code> on the same set of axes.</p><p>iv) Label the points <code class='latex inline'>\displaystyle O(0,0), A(-2,4) </code>, and <code class='latex inline'>\displaystyle B(1,1) </code> on the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} </code>. Determine the images of these points on the transformed function. Label the images <code class='latex inline'>\displaystyle O^{\prime}, A^{\prime} </code>, and <code class='latex inline'>\displaystyle B^{\prime} </code>.</p><p><code class='latex inline'>\displaystyle f(x)=-(x-2)^{2} </code></p>
<p>Describe the transformations in order that you would apply to the graph of <code class='latex inline'>y = x^2</code> to sketch each quadratic relation.</p><p><code class='latex inline'> \displaystyle y = \frac{1}{2}(x + 3)^2 - 8 </code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex at <code class='latex inline'>\displaystyle (0,0) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>translated 2 units to the right and reflected in the <code class='latex inline'>\displaystyle x </code>-axis</p>
<p>Write the equation of the quadratic function, first in vertex form and then in standard form.</p><p>vertex (1, — 7) and passing through (—2, 29)</p>
<p>For each function,</p><p>i) identify the values of the parameters <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code></p><p>ii) identify the transformations</p><p>iii) use transformations to graph the function and check that it is correct with a table of values or a graphing calculator <code class='latex inline'>\displaystyle f(x)=-3 x^{2} </code></p>
<p>Graph each parabola using a graphing calculator. </p><p><code class='latex inline'> y = (x+3)^2</code></p><p><code class='latex inline'> y = (x-4)^2</code></p><p><code class='latex inline'> y = (x+2)^2 + 5</code></p><p><code class='latex inline'> y = (x+5)^2 - 3</code></p><p><code class='latex inline'> y = (x-6)^2 +7 </code></p><p><code class='latex inline'> y = (x-1)^2 - 8</code></p><p><code class='latex inline'> y = -(x+8)^2 - 4</code></p><p><code class='latex inline'> y = 3(x+7)^2 - 2</code></p><p><code class='latex inline'> y = -2(x+3)^2 - 6</code></p><p><code class='latex inline'> y = -\dfrac{1}{2}(x+5)^2 - 3</code></p>
<p>For each of the following functions state </p><p>i) the direction of opening</p><p>ii)the coordinate of the vertex</p><p>iii) the range</p><p>iv) the equation of the axis of symmetry</p><p><code class='latex inline'>\displaystyle f(x)=\frac{1}{2}(x+3)^{2}-5 </code></p>
<p>a) The graph on the bottom resulted from transforming the green graph of <code class='latex inline'>y = x^2</code>. Determine the equation of the black graph. Explain your reasoning.</p><p>b) State the transformations that were applied to the graph of y =x^2 to result in the bottom graph.</p><img src="/qimages/6238" />
<p>Write an equation for a parabola with the given vertex and given value of <code class='latex inline'> a </code> .</p><p><code class='latex inline'>\displaystyle (0,0) ; a=5 </code></p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = -\dfrac{1}{2}(x+5)^2 - 3</code></p>
<p>Graph the function <code class='latex inline'>\displaystyle f(x)=-3 x^{2} </code>.</p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=x^{2}-11.4 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions.</p><p>opens downward, wider than <code class='latex inline'>\displaystyle y=x^{2} </code>, and vertex <code class='latex inline'>\displaystyle (-2,0) </code></p>
<p>If <code class='latex inline'>\displaystyle y=x^{2}-4 </code>, what value(s) of <code class='latex inline'>\displaystyle x </code> give each of the following values of <code class='latex inline'>\displaystyle y ? </code></p><p><code class='latex inline'>12</code></p>
<p> Write an equation of the quadratic equation that satisfies each set of conditions.</p><p>The parabola has a minimum value of <code class='latex inline'>-7</code> at <code class='latex inline'>x = 3</code>.</p>
<p> State the transformations that are applied to each parent function, resulting in the given transformed function. Sketch the graphs of the</p><p><code class='latex inline'>f(x) =x^2, y=f(x-3)+2</code></p>
<p>For each of the following, state the equation of a parabola congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> with the given property.</p><p>The graph is 2 units to the right of the graph of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p>
<p>For each graph, write the equation in both factored and standard forms.</p><img src="/qimages/163704" />
<p>Sketch the graphs below.</p><p><code class='latex inline'> \displaystyle y = \frac{2}{3}x^2 -1 </code></p>
<p>For each of the following, state the equation of a parabola congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> with the given property.</p><p>The graph is vertically compressed by a factor of 4 .</p><p>The graph is vertically compressed by a factor of 4 .</p>
<p>Use graphing technology to graph the function <code class='latex inline'>f(x) = (x + 3)^{2} - 1</code>.</p><p>a) Determine the invariant point(s), if any, when <code class='latex inline'>f(x)</code> is reflected in </p> <ul> <li>i) the <em>x</em>-axis</li> <li>ii) the <em>y</em>-axis</li> <li>iii) the <em>x</em>-axis and the <em>y</em>-axis</li> </ul> <p>b) Write the equation of a quadratic function that has an invariant point for all the reflections in part a). Justify your answer. </p>
<p>Sketch the graph of the parabola. Describe the transformation from the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'> \displaystyle y = (x - 2)^2 </code></p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=x^{2}+5 </code></p>
<ol> <li>Find <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> so that a parabola <code class='latex inline'>\displaystyle y=a x^{2}+k </code> passes through each pair of points.</li> </ol> <p><code class='latex inline'>\displaystyle (3,-10) </code> and <code class='latex inline'>\displaystyle (-1,-2) </code></p>
<p>What is the <code class='latex inline'>\displaystyle y </code> -value of the <code class='latex inline'>\displaystyle y </code> -intercept of the quadratic function <code class='latex inline'>\displaystyle y=2(x+2)^{2}-5 ? </code></p>
<p>Reasoning Is <code class='latex inline'>\displaystyle y=0(x-4)^{2}+3 </code> a quadratic function? Explain.</p>
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=2 x^{2} ; </code> minimum value <code class='latex inline'>\displaystyle -6 </code>; axis of symmetry <code class='latex inline'>\displaystyle x=-5 </code></p>
<p>Determine the answers to the following questions for each of the given transformed quadratic functions.</p><p>i) How does the shape of the graph compare with the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} ? </code></p><p>ii) What are the coordinates of the vertex and the equation of the axis of symmetry?</p><p>iii) Graph the transformed function and <code class='latex inline'>\displaystyle f(x)=x^{2} </code> on the same set of axes.</p><p>iv) Label the points <code class='latex inline'>\displaystyle O(0,0), A(-2,4) </code>, and <code class='latex inline'>\displaystyle B(1,1) </code> on the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} </code>. Determine the images of these points on the transformed function. Label the images <code class='latex inline'>\displaystyle O^{\prime}, A^{\prime} </code>, and <code class='latex inline'>\displaystyle B^{\prime} </code>.</p><p><code class='latex inline'>\displaystyle f(x)=(x+2)^{2}-2 </code></p>
<p>Use the Quadratic Formula to prove each statement.</p><p>a. The sum of the solutions of the quadratic equation <code class='latex inline'>\displaystyle a x^{2}+b x+c=0 </code> is <code class='latex inline'>\displaystyle -\frac{b}{a} </code>.</p><p>b. The product of the solutions of the quadratic equation <code class='latex inline'>\displaystyle a x^{2}+b x+c=0 </code> is <code class='latex inline'>\displaystyle \frac{c}{a} </code>.</p>
<ol> <li>Communication Graph <code class='latex inline'>\displaystyle x+y=4 </code> and <code class='latex inline'>\displaystyle y=x^{2}-2 </code> on the same axes. a) What are the coordinates of the intersection points? b) Describe how you found the intersection points.</li> </ol>
<p>For each function, determine the vertex and two points that satisfy the equation. Use the information to sketch the graph of each.</p><p><code class='latex inline'> \displaystyle y = 2(x - 1)^2 -3 </code></p>
<p>If <code class='latex inline'>\displaystyle y=x^{2}-4 </code>, what value(s) of <code class='latex inline'>\displaystyle x </code> give each of the following values of <code class='latex inline'>\displaystyle y ? </code></p><p><code class='latex inline'>7</code></p>
<img src="/qimages/2493" /><p>Which equation represents the graph shown? Explain your reasoning.</p><p><strong>a)</strong> <code class='latex inline'>\displaystyle{y=-\frac{2}{3}x^2+5}</code></p><p><strong>b)</strong> <code class='latex inline'>\displaystyle{y=-(x-3)^2+5}</code></p><p><strong>c)</strong> <code class='latex inline'>\displaystyle{y=-\frac{2}{3}(x-3)^2+5}</code></p><p><strong>d)</strong> <code class='latex inline'>\displaystyle{y=\frac{2}{3}(x-3)^2+5}</code></p>
<p>Which equation describes the black graph?</p><img src="/qimages/145265" /><p><code class='latex inline'>\displaystyle \begin{array}{ll}\text { A. } y=0.2 x^{2} & \text { C. } y=-0.2 x^{2} \\ \text { B. } y=-2 x^{2} & \text { D. } y=2 x^{2}\end{array} </code></p>
<p>Write an equation for a parabola with the given vertex and given value of <code class='latex inline'> a </code> .</p><p><code class='latex inline'>\displaystyle (0,3) ; a=0.2 </code></p>
<p>Sketch the graph of the quadratic relation. Describe the transformation from the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'>\displaystyle y =x^2 + 2 </code></p>
<p>State the following for the given parabola.</p> <ul> <li>Vertex</li> <li>Axis of symmetry</li> <li>Stretch or compression factor relative to <code class='latex inline'>y=x^2</code></li> <li>Direction of opening</li> <li>Values x may take</li> <li>Values y may take</li> </ul> <p><code class='latex inline'>\displaystyle y =(x - 1)^2 -4 </code></p>
<p>Determine the zeros of each function.</p><p><code class='latex inline'> \displaystyle g(x) = -3(x - 7)^2 - 121 </code></p>
<p>Describe the transformations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain a graph of each quadratic relation. Sketch the graph by hand. Start with the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> and use the appropriate transformations.</p><p><code class='latex inline'>\displaystyle y=-(x-3)^{2}+2 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions. </p><p>opens downward, congruent with <code class='latex inline'>\displaystyle y=\frac{1}{2} x^{2} </code>, vertex <code class='latex inline'>\displaystyle (-3,2) </code></p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-2.9 x^{2}-8.3 </code></p>
<p>The parabola shown is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code></p><p>a) What are the zeros of the function?</p><p>b) Write the equation in factored form.</p><img src="/qimages/163664" />
<p>Use the point marked on each parabola, as well as the vertex of the parabola, to determine the equation of the parabola in vertex form.</p><img src="/qimages/6234" />
<p>Transformations are applied to the graphs of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain the black parabolas. Describe the transformations that were applied. Write an equation for each black parabola.</p><img src="/qimages/163448" />
<p>State the vertex, axis of symmetry, maximum or minimum value, domain, and range of each function. Then graph each function.</p><p><code class='latex inline'>\displaystyle f(x)=-(x+5)^{2}-7 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions. </p><p>vertex <code class='latex inline'>\displaystyle (4,5) </code>, passes through <code class='latex inline'>\displaystyle (2,9) </code></p>
<p>The vertex of a parabola is at <code class='latex inline'>\displaystyle (2,5) </code>, and the parabola passes through <code class='latex inline'>\displaystyle (4,-1) </code>. What is the equation of the parabola?</p><p>A. <code class='latex inline'>\displaystyle y=-2.5(x-2)^{2}+5 </code></p><p>B. <code class='latex inline'>\displaystyle y=-2.5(x+2)^{2}+5 </code></p><p>C. <code class='latex inline'>\displaystyle y=-1.5(x-5)^{2}+2 </code></p><p>D. <code class='latex inline'>\displaystyle y=-1.5(x-2)^{2}+5 </code></p>
<p>Determine an equation to represent each parabola.</p><img src="/qimages/5555" />
<ol> <li>Communication If a function of the form <code class='latex inline'>\displaystyle y=a x^{2}+k </code> has an <code class='latex inline'>\displaystyle x </code> -intercept of <code class='latex inline'>\displaystyle 7.5 </code>, what is the other <code class='latex inline'>\displaystyle x </code> -intercept? Explain how you know.</li> </ol>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph `$y=x^2$ is reflected in the x-axis and then stretched vertically by a factor of 6. </p>
<ol> <li>Write an equation for the parabola with the given vertex and the given value of <code class='latex inline'>\displaystyle a </code>.</li> </ol> <p><code class='latex inline'>\displaystyle (-5,0) ; a=-1 </code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex at <code class='latex inline'>\displaystyle (0,0) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>compressed vertically by a factor of 2</p>
<p>For each of the following, state the equation of a parabola congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> with the given property.</p><p>The graph is vertically stretched by a factor of 3 and is 2 units to the right and 1 unit down from the graph of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-x^{2}-1 </code></p>
<ol> <li>Communication Describe what happens to the point <code class='latex inline'>\displaystyle (2,4) </code> on the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> when each pair of transformations is applied to the parabola in the given order.</li> </ol> <p>a reflection in the <code class='latex inline'>\displaystyle x </code> -axis, followed by a vertical translation of 3</p>
<p>For each function, determine the vertex and two points that satisfy the equation. Use the information to sketch the graph of each.</p><p><code class='latex inline'> \displaystyle y = x^2 + 4 </code></p>
<p>Determine the maximum or minimum value for each.</p><p>(a) <code class='latex inline'> \displaystyle y = -4(x + 1)^2 + 6 </code></p><p>(b) <code class='latex inline'> \displaystyle f(x) = (x - 5)^2 </code></p>
<p>Sketch the graph of the parabola. Describe the transformation from the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'> \displaystyle y = -2x^2 </code></p>
<p>Describe the transformations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain the graph of each quadratic relation.</p><p><code class='latex inline'>\displaystyle y=-3 x^{2}-4 </code></p>
<p>A parabola <code class='latex inline'>y = ax^2 + k</code> passes through the points <code class='latex inline'>(-1, 3)</code> and <code class='latex inline'>(3, -13)</code>. Find the values of <code class='latex inline'>a</code> and <code class='latex inline'>k</code>.</p>
<p>Sketch the parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.</p><p><code class='latex inline'>\displaystyle y = (x+ 1)^2 -1 </code></p>
<p>Write the equations of two different quadratic relations that match each description.</p><p>The graph has a narrower opening than the graph of <code class='latex inline'>y=2x^2</code></p>
<p>Match each graph with the appropriate equation.</p><img src="/qimages/22336" /><p>a) <code class='latex inline'>y=-(x-2)^2+3</code></p><p>b) <code class='latex inline'>y=-(x-1)^2+4</code></p><p>c) <code class='latex inline'>y=-(x+3)^2-2</code></p><p>d) <code class='latex inline'>y=-(x+2)^2-1</code></p>
<p>Identify the transformation(s) that must be applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to create a graph of each equation. Then state the coordinates of the image of the point <code class='latex inline'>\displaystyle (2,4) </code>.</p><p><code class='latex inline'>\displaystyle y=-5 x^{2} </code></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph <code class='latex inline'>y=x^2</code> is stretched vertically by a factor of 5. </p>
<p>Write the equation of a parabola with each set of properties.</p> <ul> <li>vertex at <code class='latex inline'>(0,4)</code>, opens upward, the same shape as <code class='latex inline'>y=x^2</code></li> </ul>
<p>State the vertex, axis of symmetry, maximum or minimum value, domain, and range of each function. Then graph each function.</p><p><code class='latex inline'>\displaystyle f(x)=(x-3)^{2}+6 </code></p>
<p>Express each quadratic function in factored form. Then determine the zeros, the equation of the axis of symmetry, and the coordinates of the vertex.</p><p><code class='latex inline'>\displaystyle f(x)=-x^{2}+100 </code></p>
<p>Graph each parabola. State the coordinates of the vertex. Find any intercepts.</p><p><code class='latex inline'>\displaystyle y=16+x^{2} </code></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph <code class='latex inline'>y=x^2</code> is compressed vertically by a factor of <code class='latex inline'>\dfrac{1}{6}</code>.</p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph <code class='latex inline'>y=x^2</code> is compressed vertically by a factor of <code class='latex inline'>\dfrac{1}{4}</code></p>
<p>Use the graph of each quadratic relation to determine the roots to each quadratic equation, where <code class='latex inline'>y=0</code>.</p><img src="/qimages/855" />
<p>Examine each parabola.</p> <ul> <li>i) Determine the coordinates of the vertex.</li> <li>ii) Determine the zeros.</li> <li>iii) Determine the equation of the axis of symmetry.</li> <li>iv) If you calculated the second differences, would they be positive or negative? Explain.</li> </ul> <img src="/qimages/758" />
<p>For each graph, state the y-intercept, the zeros, the coordinates of the vertex, and the equation of the axis of symmetry.</p><img src="/qimages/756" />
<p>State two different ways to determine the number of zeros of the function <code class='latex inline'>f(x)=2(x+1)^2-6</code></p>
<p>For each of the following, state the equation of a parabola congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> with the given property.</p><p>The graph is 4 units to the left of the graph of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p>
<ol> <li>Systems of equations a) Graph <code class='latex inline'>\displaystyle y=3 x+3 </code> and <code class='latex inline'>\displaystyle y=(x+2)^{2}-3 </code> on the same set of axes.</li> </ol> <p>b) What are the coordinates of the intersection points? c) Communication Describe how you found the intersection points.</p>
<p>Graph each function. Describe how it was translated from <code class='latex inline'>f(x) = x^2</code>.</p><p><code class='latex inline'>\displaystyle f(x)=(x-2)^{2} </code></p>
<p>Determine the y-intercept, zeros, equation of the axis of symmetry, and vertex of each quadratic relation. Then sketch its graph.</p><p><code class='latex inline'>\displaystyle y = -0.5(x+4)^2 </code></p>
<p>Determine an algebraic expression for the solution, <code class='latex inline'>x</code>, to the equation <code class='latex inline'>0=a(x-h)^2+k</code>. Do not expand the equation.</p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y=x^2</code> is translated 7 units to the right. </p>
<p>For each graph, write the equation in both factored and standard forms.</p><img src="/qimages/163705" />
<p>Determine the zeros and the maximum or minimum value for each function.</p><p><code class='latex inline'>\displaystyle f(x)=-x^{2}+49 </code></p>
<p>Describe the translations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain a graph of each quadratic function. Sketch the graph. <code class='latex inline'>\displaystyle y=3 x^{2}-4 </code></p>
<p>For each quadratic relation,</p> <ol> <li>state the stretch/compression factor and the horizontal/vertical translations</li> <li>determine whether the graph is reflected m‘ the x—axis</li> <li>state the vertex and the equation of the axis of symmetry</li> </ol> <p><code class='latex inline'> \displaystyle y = -\frac{1}{2}(x+ 4)^2 </code></p>
<ol> <li>Write an equation for the parabola with the given vertex and the given value of <code class='latex inline'>\displaystyle a </code>.</li> </ol> <p><code class='latex inline'>\displaystyle (3,-5) ; a=2 </code></p>
<p>Find </p> <ul> <li>Vertex</li> <li>Axis of Symmetry</li> <li>Determine if it opens up/down</li> <li>Range</li> <li>Sketch</li> </ul> <p><code class='latex inline'> \displaystyle f(x) = \frac{1}{2}(x - 4)^2 + 3 </code></p>
<p>A parabola has equation <code class='latex inline'>y=(x +2)^2</code></p><p><strong>(a)</strong> Write its <code class='latex inline'>x</code>-intercepts.</p><p><strong>(b)</strong> Determine the coordinates of its vertex.</p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=2+x^{2} </code></p>
<p>Identify the transformation(s) that must be applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to create a graph of each equation. Then state the coordinates of the image of the point <code class='latex inline'>\displaystyle (2,4) </code>.</p><p><code class='latex inline'>\displaystyle y=0.25 x^{2} </code></p>
<p>Sketch the graphs below.</p><p><code class='latex inline'> \displaystyle y = \frac{1}{2}(x + 3)^2 - 8 </code></p>
<p>State the maximum or minimum value of each relation.</p><p>a) <img src="/qimages/754" /></p><p>b) <img src="/qimages/755" /></p><p>c) <img src="/qimages/756" /></p>
<ol> <li>Write an equation for the parabola with the given vertex and the given value of <code class='latex inline'>\displaystyle a </code>.</li> </ol> <p><code class='latex inline'>\displaystyle (6,7) ; a=-3 </code></p>
<p>Without graphing, determine the number of x-intercepts that the relation has.</p><p><code class='latex inline'>\displaystyle y = -1.8(x- 3)^2 + 2 </code></p>
<p>The following transformations are applied to a parabola with the equation <code class='latex inline'>y=2(x+3)^2-1</code> Determine the equation that will result after each transformation.</p> <ul> <li>a compression by a factor of <code class='latex inline'>\displaystyle{\frac{1}{4}}</code>, followed by a reflection in the y-axis.</li> </ul>
<p>Write the equations of two different quadratic relations that match each description.</p><p>a) The graph is narrower than the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> near its vertex.</p><p>b) The graph is wider than the graph of <code class='latex inline'>\displaystyle y=-x^{2} </code> near its vertex.</p><p>c) The graph opens downward and is narrower than the graph of <code class='latex inline'>\displaystyle y=3 x^{2} </code> near its vertex.</p>
<p>The parabola <code class='latex inline'>y = x^2</code> is transformed in two different ways to produce the parabolas <code class='latex inline'>y=2(x-4)^2+5 </code> and <code class='latex inline'>y =2(x-5)^2+4</code>. How are these transformations the same, and how are they different?</p>
<p>Determine the equation of the parabola with vertex <code class='latex inline'>(4, -5)</code> and that passes through <code class='latex inline'>(-1, -3)</code></p>
<p>State the zeros, vertex, and equation of the axis of symmetry of the parabola at the right.</p><img src="/qimages/9866" />
<p>Find each value if <code class='latex inline'>\displaystyle f(x)=3 x+2, g(x)=-2 x^{2} </code>, and <code class='latex inline'>\displaystyle h(x)=-4 x^{2}-2 x+5 </code></p><p><code class='latex inline'>\displaystyle f(-5) </code></p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-4 x^{2} </code></p>
<p>Write the equation of a parabola with each set of properties.</p> <ul> <li>vertex at <code class='latex inline'>(2,-3)</code>, opens upward, narrower than <code class='latex inline'>y=x^2</code></li> </ul>
<p>Write the equation of a parabola with each set of properties.</p> <ul> <li>vertex at <code class='latex inline'>(5,0)</code>, opens downward, the same shape as <code class='latex inline'>y=x^2</code></li> </ul>
<p>A graphing calculator was used together with the vertex form <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> to graph the screens shown. For the set of graphs on each screen, tell which of the variables <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code> remained constant and which changed. Give possible values for the variables that remained constant.</p><img src="/qimages/163482" />
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=2 x^{2} </code>; maximum at <code class='latex inline'>\displaystyle (2,-3) </code></p>
<p>Write an equation for the parabola with vertex (-1,-7), opening upward, and with a vertical compression of factor 0.4.</p>
<p>Determine any x-intercepts, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle y=8-4 x^{2} </code></p>
<p>Write the equation of a parabola with each set of properties.</p> <ul> <li>vertex at <code class='latex inline'>(3,-4)</code>, no zeros, wider than <code class='latex inline'>y=x^2</code></li> </ul>
<p>State the number of times that each relation passes through the x-axis. Justify your answer</p><p> <code class='latex inline'>y=-2(x+5)^2-8</code></p>
<p>Determine the answers to the following questions for each of the given transformed quadratic functions.</p><p>i) How does the shape of the graph compare with the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} ? </code></p><p>ii) What are the coordinates of the vertex and the equation of the axis of symmetry?</p><p>iii) Graph the transformed function and <code class='latex inline'>\displaystyle f(x)=x^{2} </code> on the same set of axes.</p><p>iv) Label the points <code class='latex inline'>\displaystyle O(0,0), A(-2,4) </code>, and <code class='latex inline'>\displaystyle B(1,1) </code> on the graph of <code class='latex inline'>\displaystyle f(x)=x^{2} </code>. Determine the images of these points on the transformed function. Label the images <code class='latex inline'>\displaystyle O^{\prime}, A^{\prime} </code>, and <code class='latex inline'>\displaystyle B^{\prime} </code>.</p><p><code class='latex inline'>\displaystyle f(x)=\frac{1}{2} x^{2}+2 </code></p>
<p>Match each function to its graph.</p><p>a) <code class='latex inline'>\displaystyle f(x)=(x+3)^{2}+1 </code></p><p>c) <code class='latex inline'>\displaystyle f(x)=-(x-3)^{2}-2 </code></p><p>b) <code class='latex inline'>\displaystyle f(x)=-2(x+4)^{2}+3 </code></p><p>d) <code class='latex inline'>\displaystyle f(x)=\frac{1}{2}(x-3)^{2}-5 </code></p><img src="/qimages/163444" />
<p>Write an equation for each parabola. </p><img src="/qimages/22261" />
<p>Find <code class='latex inline'> a </code> and <code class='latex inline'> k </code> so that the given points lie on the parabola</p><p><code class='latex inline'>\displaystyle y=a(x+3)^{2}+k ;(-5,-8),(1,-20) </code></p>
<p>Transformations are applied to the graphs of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain the black parabolas. Describe the transformations that were applied. Write an equation for each black parabola.</p><img src="/qimages/163446" />
<p>a) Describe the transformations to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain <code class='latex inline'>\displaystyle y=-2(x+5)^{2}-3 </code></p><p>b) Graph <code class='latex inline'>\displaystyle y=x^{2} </code>. Then apply the transformations in part (a) to graph <code class='latex inline'>\displaystyle y=-2(x+5)^{2}-3 </code></p>
<p>Find </p> <ul> <li>i) Vertex</li> <li>ii) Axis of Symmetry</li> <li>iii) Determine if it opens up/down</li> <li>iv) Range</li> <li>v) Sketch</li> </ul> <p><code class='latex inline'> \displaystyle f(x) = (x - 3)^2 + 1 </code></p>
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=5 x^{2} </code>; minimum at <code class='latex inline'>\displaystyle (4.5,0) </code></p>
<p>For the graph, write the quadratic equation in vertex form.</p><img src="/qimages/2081" />
<p>Write the equation of the parabola in vertex form.</p><img src="/qimages/1543" />
<p>Write in standard form the equation of the parabola passing through the given points.</p><p><code class='latex inline'>\displaystyle (3,4),(-2,9),(2,1) </code></p>
<p>Find </p> <ul> <li>Vertex</li> <li>Axis of Symmetry</li> <li>Determine if it opens up/down</li> <li>Range</li> <li>Sketch</li> </ul> <p><code class='latex inline'> \displaystyle f(x) = -(x + 1)^2 - 5 </code></p>
<p>Sketch the graphs below.</p><p><code class='latex inline'> \displaystyle y = 4(x -2)^2 - 5 </code></p>
<ol> <li>Write an equation for the parabola with the given vertex and the given value of <code class='latex inline'>\displaystyle a </code>.</li> </ol> <p><code class='latex inline'>\displaystyle (-1,-1) ; a=-0.5 </code></p>
<p>Determine the zeros of each function.</p><p><code class='latex inline'> \displaystyle f(x) = (x - 3)^2 - 121 </code></p>
<p>Sketch each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</p><p><code class='latex inline'>\displaystyle y=-(x-2)^{2}-5 </code></p>
<p>State the following for the given parabola.</p> <ul> <li>Vertex</li> <li>Axis of symmetry</li> <li>Stretch or compression factor relative to <code class='latex inline'>y=x^2</code></li> <li>Direction of opening</li> <li>Values x may take</li> <li>Values y may take</li> </ul> <p><code class='latex inline'>\displaystyle y =2(x + 3)^2 + 1 </code></p>
<p>List the sequence of steps required to graph each function.</p><p><code class='latex inline'>\displaystyle f(x)=3(x+2)^{2} </code></p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle y=2 x^{2} </code></p>
<p>The graph shows <code class='latex inline'>\displaystyle f(x)=2(x-3)^{2}-1 </code>.</p><p>a) Evaluate <code class='latex inline'>\displaystyle f(0) </code>.</p><p>b) What does <code class='latex inline'>\displaystyle f(0) </code> represent on the graph of <code class='latex inline'>\displaystyle f </code> ?</p><p>c) If <code class='latex inline'>\displaystyle f(x)=6 </code>, determine possible values of <code class='latex inline'>\displaystyle x </code>.</p><p>d) Does <code class='latex inline'>\displaystyle f(3)=4 </code> for this function? Explain.</p><img src="/qimages/26095" />
<p>For each of the following, state the condition on <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> such that the parabola <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> has the given property.</p><p>The parabola intersects the <code class='latex inline'>\displaystyle x </code>-axis at one point.</p>
<p>Write each function in standard form.</p><p><code class='latex inline'>\displaystyle g(x)=-(x-3)^{2}-8 </code></p>
<p>Sketch the parabola. Label the coordinates of the vertex and the equation of the axis of symmetry.</p><p><code class='latex inline'>\displaystyle y = x^2 -3 </code></p>
<p>Match each graph with the correct equation. The graph of <code class='latex inline'>\displaystyle y=x^{2} </code> is shown in green in each diagram.</p><p>a) <code class='latex inline'>\displaystyle y=x^{2}+5 </code></p><p>c) <code class='latex inline'>\displaystyle y=-2 x^{2}+5 </code></p><p>b) <code class='latex inline'>\displaystyle y=(x+5)^{2} </code></p><p>d) <code class='latex inline'>\displaystyle y=2(x+5)^{2} </code></p><img src="/qimages/163393" />
<p>State the following for the given parabola.</p> <ul> <li>Vertex</li> <li>Axis of symmetry</li> <li>Stretch or compression factor relative to <code class='latex inline'>y=x^2</code></li> <li>Direction of opening</li> <li>Values x may take</li> <li>Values y may take</li> </ul> <p><code class='latex inline'>\displaystyle y =\frac{1}{4}(x - 5)^2 + 1 </code></p>
<p>Explain why it makes sense that each statement about the graph of <code class='latex inline'>\displaystyle y=a x^{2} </code> is true.</p><p>a) If <code class='latex inline'>\displaystyle a < 0 </code>, then the parabola opens downward.</p><p>b) If <code class='latex inline'>\displaystyle a </code> is a rational number between <code class='latex inline'>\displaystyle -1 </code> and 1 , then the parabola is wider than the graph of <code class='latex inline'>\displaystyle y=x^{2} </code></p><p>c) The vertex is always <code class='latex inline'>\displaystyle (0,0) </code>.</p>
<p>Describe the transformations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain a graph of each quadratic relation. Sketch the graph by hand. Start with the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> and use the appropriate transformations.</p><p><code class='latex inline'>\displaystyle y=2(x+1)^{2}-8 </code></p>
<p>Graph each function. Describe how it was translated from <code class='latex inline'>f(x) = x^2</code>.</p><p><code class='latex inline'>\displaystyle f(x)=x^{2}+3 </code></p>
<p>State the number of times that each relation passes through the x-axis. Justify your answer</p><p><code class='latex inline'>y=5(x-12)^2+81</code></p>
<p>Write an equation for the parabola with vertex at <code class='latex inline'>(4, -1)</code>, opening upward, and with a vertical compression of factor 0.3.</p>
<p>Determine the equations of three different parabolas with a vertex at <code class='latex inline'>(-2,3)</code>. Describe how the graphs of the parabolas are different from each other. Then sketch the graphs of the three relations on the same set of axes.</p>
<p>Express each quadratic function in factored form. Then determine the zeros, the equation of the axis of symmetry, and the coordinates of the vertex.</p><p><code class='latex inline'>\displaystyle f(x)=2 x^{2}+12 x </code></p>
<p>The vertex of a parabola <code class='latex inline'>(-2, -4)</code>. One x-intercept is 7. What is the other x-intercept?</p>
<p>State the following for the given parabola.</p> <ul> <li>Vertex</li> <li>Axis of symmetry</li> <li>Stretch or compression factor relative to <code class='latex inline'>y=x^2</code></li> <li>Direction of opening</li> <li>Values x may take</li> <li>Values y may take</li> </ul> <p><code class='latex inline'>\displaystyle y = -(x + 2)^2 + 6 </code></p>
<ol> <li>Communication Describe what happens to the point <code class='latex inline'>\displaystyle (2,4) </code> on the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> when each pair of transformations is applied to the parabola in the given order.</li> </ol> <p>a vertical translation of <code class='latex inline'>\displaystyle -2 </code>, followed by a reflection in the <code class='latex inline'>\displaystyle x </code> -axis</p>
<p>Transformations are applied to the graphs of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain the black parabolas. Describe the transformations that were applied. Write an equation for each black parabola.</p><img src="/qimages/163449" />
<p>The vertex of a quadratic relation is (4,-12).</p><p><strong>a)</strong> Write an equation to describe all parabolas with this vertex.</p><p><strong>b)</strong> A parabola with the given vertex passes through point (13,15). Determine the value of <code class='latex inline'>a</code> for this parabola.</p><p><strong>c)</strong> Write the equation of the relation for part b).</p><p><strong>d)</strong> State the transformations that must be applied to <code class='latex inline'>y=x^2</code> to obtain the quadratic relation you wrote for part c).</p><p><strong>e)</strong> Graph the quadratic relation you wrote for part c).</p>
<p>Write one sentence that compares each pair of graphs. </p><p><code class='latex inline'>\displaystyle y=-x^{2} </code> and <code class='latex inline'>\displaystyle y=-\frac{1}{3} x^{2} </code></p>
<p>Write an equation for each parabola. </p><img src="/qimages/22262" />
<p>Write an equation for the parabola with the given vertex and passing through the given point.</p><p>vertex <code class='latex inline'> (-4,-5) ; </code> point <code class='latex inline'> (-2,-1) </code> </p>
<p>a) Describe how the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> can be transformed into the graph of the given quadratic function.</p><p>ii) <code class='latex inline'>\displaystyle y=\frac{1}{4}(x-5)^{2} </code></p><p>b) List the domain and range of each function. Compare these with the original graph of <code class='latex inline'>\displaystyle y=x^{2} </code></p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle y=-7 x^{2} </code></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y=x^2</code> is translated 6 units upward. </p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = (x-6)^2 +7 </code></p>
<p>a) Describe how the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> can be transformed into the graph of the given quadratic function.</p><p>iii) <code class='latex inline'>\displaystyle y=-3(x+5)^{2}-7 </code></p><p>b) List the domain and range of each function. Compare these with the original graph of <code class='latex inline'>\displaystyle y=x^{2} </code></p>
<p>Sketch the graphs of each of the following: <code class='latex inline'>\displaystyle y=x^{2}-2 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions.</p><p>opens downward, congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, and vertex <code class='latex inline'>\displaystyle (5,0) </code></p>
<p>For each function,</p><p>i) identify the values of the parameters <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code></p><p>ii) identify the transformations</p><p>iii) use transformations to graph the function and check that it is correct with a table of values or a graphing calculator <code class='latex inline'>\displaystyle f(x)=-(x-2)^{2} </code></p>
<p>For each graph, state the y-intercept, the zeros, the coordinates of the vertex, and the equation of the axis of symmetry.</p><p> <img src="/qimages/754" /></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y=x^2</code> is translated 8 units downward. </p>
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=x^{2} ; </code> opens down; vertex at <code class='latex inline'>\displaystyle (-3,0) </code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, opens upward, and has vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Now find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>translated 3 units up</p>
<p>Write an equation for each parabola. </p><img src="/qimages/22260" />
<p>Write the equation of the parabola in vertex form.</p><img src="/qimages/1544" />
<p>Predict what the graphs of each group of equations would look like. Check your predictions by using graphing technology.</p><p>a) <code class='latex inline'>\displaystyle \begin{array}{lll}f(x)=10 x^{2} & \text { b) } f(x) & =0.1 x^{2} \\ f(x)=100 x^{2} & f(x) & =0.01 x^{2} \\ f(x)=1000 x^{2} & f(x)=0.001 x^{2} \\ f(x)=10000 x^{2} & f(x)=0.0001 x^{2}\end{array} </code></p>
<p>The following transformations are applied to a parabola with the equation <code class='latex inline'>y=2(x+3)^2-1</code> Determine the equation that will result after each transformation.</p> <ul> <li>a reflection in the <code class='latex inline'>x-</code>axis, followed by a translation 5 units down</li> </ul>
<p>For each quadratic relation, state</p> <ul> <li>i) the equation of the axis of symmetry</li> <li>ii) the coordinates of the vertex</li> <li>iii) the y-intercept</li> <li>iv) the zeros</li> <li>v) the maximum or minimum value</li> </ul> <p> <img src="/qimages/760" /></p>
<ol> <li>Determine the value of <code class='latex inline'>\displaystyle k </code> so that the graph of <code class='latex inline'>\displaystyle y=(x+3)^{2}+k </code> passes through the point <code class='latex inline'>\displaystyle (1,20) </code>.</li> </ol>
<p>Describe how the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> can be transformed to the graphs of the relations from question 6 .</p>
<p>a) Describe how the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> can be transformed into the graph of the given quadratic function.</p><p>i) <code class='latex inline'>\displaystyle y=5 x^{2}-4 </code></p><p>b) List the domain and range of each function. Compare these with the original graph of <code class='latex inline'>\displaystyle y=x^{2} </code></p>
<p>Sketch each parabola and estimate any intercepts.</p><p><code class='latex inline'>\displaystyle y=(x+2)^{2}-9 </code></p>
<p>Determine the equation of the parabola. Express your answer in standard form.</p><img src="/qimages/4003" />
<p>Match each equation with the correct graph.</p><p><strong>a)</strong> <code class='latex inline'>y=2x^2-8</code></p><p><strong>b)</strong> <code class='latex inline'>y=(x+3)^2</code></p><p><strong>c)</strong> <code class='latex inline'>y=-2(x-4)^2+8</code></p><p><strong>d)</strong> <code class='latex inline'>y=(x-3)^2-8</code></p><img src="/qimages/1540" />
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = -2(x+3)^2 - 6</code></p>
<p>Sketch the graph of each equation by correctly applying the required transformation(s) to points on the graph of <code class='latex inline'>y = x^2</code>. Use a separate grid for each graph.</p><p><code class='latex inline'> \displaystyle y =-3x^2 </code></p>
<p>Determine the maximum or minimum value for each quadratic function.</p><p><code class='latex inline'>\displaystyle g(x)=-x^{2}+25 </code></p>
<p>Transformations are applied to the graphs of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain the black parabolas. Describe the transformations that were applied. Write an equation for each black parabola.</p><img src="/qimages/163447" />
<p>Describe the transformation(s) that were applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain each black graph. Write the equation of the black graph.</p><img src="/qimages/145220" />
<p>Sketch each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</p><p><code class='latex inline'>\displaystyle y=-(x+1)^{2} </code></p>
<p>Use transformations to determine the vertex, axis of symmetry, and direction of opening of each parabola. Sketch the graph.</p><p><code class='latex inline'>\displaystyle y=2(x-5)^{2} </code></p>
<ol> <li>Sketch each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=(x+5)^{2} </code></p>
<ol> <li>Find <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> so that a parabola <code class='latex inline'>\displaystyle y=a x^{2}+k </code> passes through each pair of points.</li> </ol> <p><code class='latex inline'>\displaystyle (1,-1) </code> and <code class='latex inline'>\displaystyle (2,5) </code></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y=x^2</code> is translated 6 units to the left. </p>
<p>Find an equation for the parabola with vertex (-1,3) and <em>x</em>-intercept 1. </p>
<p>Write the equation of the parabola in vertex form.</p><img src="/qimages/1542" />
<p> Determine the y-intercept, zeros, equation of the axis of symmetry, and vertex of each quadratic relation. </p><p><code class='latex inline'>y = -4(x -4)^2</code></p>
<p>Determine the values of <code class='latex inline'>h</code> and <code class='latex inline'>k</code> for each ofthe following transformations. Write the equation in the form <code class='latex inline'>y = (x - b)^2 + k</code>. Sketch the graph.</p><p>The parabola moves 3 units down and 2 units right.</p>
<p>Determine the equation of the parabola with vertex <code class='latex inline'>(4, 0)</code> and that passes through <code class='latex inline'>(11, 8)</code></p>
<ol> <li>Write an equation for the parabola with the given vertex and the given value of <code class='latex inline'>\displaystyle a </code>.</li> </ol> <p><code class='latex inline'>\displaystyle (-8,9) ; a=1.5 </code></p>
<p>Sketch the graph of the quadratic relation. Describe the transformation from the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'>\displaystyle y = (x + 3)^2 </code></p>
<p>The vertex of a quadratic relation is (4,-12).</p><p><strong>a)</strong> Write an equation to describe all parabolas with this vertex.</p><p><strong>b)</strong> A parabola with the given vertex passes through point (13,15). Determine the value of <code class='latex inline'>a</code> for this parabola.</p><p><strong>c)</strong> Write the equation of the relation for part b).</p><p><strong>d)</strong> State the transformations that must be applied to <code class='latex inline'>y=x^2</code> to obtain the quadratic relation you wrote for part c).</p>
<p>a. Let <code class='latex inline'>\displaystyle a > 0 </code>. Use algebraic or arithmetic ideas to explain why the lowest point on the graph of <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> must occur when <code class='latex inline'>\displaystyle x=h . </code></p><p>b. Suppose that the function in part <code class='latex inline'>\displaystyle (a) </code> is <code class='latex inline'>\displaystyle y=a(x-h)^{3}+k . </code> Is your reasoning still valid? Explain.</p>
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=x^{2} ; </code> opens up; vertex at <code class='latex inline'>\displaystyle (1,5) </code></p>
<p>For the graph, write the quadratic equation in vertex form.</p><img src="/qimages/2082" />
<p>For each function,</p><p>i) identify the values of the parameters <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code></p><p>ii) identify the transformations</p><p>iii) use transformations to graph the function and check that it is correct with a table of values or a graphing calculator <code class='latex inline'>\displaystyle f(x)=\frac{1}{2}(x+3)^{2} </code></p>
<img src="/qimages/63978" /> <ol> <li>The four graphs represent the four equations <code class='latex inline'>\displaystyle y=3(x-1)^{2}+2, y=3(x+1)^{2}-2, y=-3(x+1)^{2}+2 </code> and <code class='latex inline'>\displaystyle y=-3(x-1)^{2}-2 . </code> Match each graph with the correct equation.</li> </ol>
<p>For each function,</p><p>i) identify the values of the parameters <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code></p><p>ii) identify the transformations</p><p>iii) use transformations to graph the function and check that it is correct with a table of values or a graphing calculator <code class='latex inline'>\displaystyle f(x)=-x^{2}-2 </code></p>
<p>Identify the transformation(s) that must be applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to create a graph of each equation. Then state the coordinates of the image of the point <code class='latex inline'>\displaystyle (2,4) </code>.</p><p><code class='latex inline'>\displaystyle y=-x^{2} </code></p>
<p>Describe the translations applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> to obtain a graph of each quadratic function. Sketch the graph. <code class='latex inline'>\displaystyle y=-\frac{1}{3}(x+4)^{2} </code></p>
<p>For each of the following, state the condition on <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> such that the parabola <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> has the given property.</p><p> The parabola intersects the <code class='latex inline'>\displaystyle x </code>-axis at two distinct points.</p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is translated 4 units to the left</p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle f(x)=3 \frac{2}{5} x^{2} </code></p>
<p>The graph of <code class='latex inline'>y = x^2</code> is horizontally stretched by a factor of 2, reflected in the x—axis, and shifted 3 units down. Find the equation that results from the transformation, and graph it.</p>
<p>For each function below, identify the equation of the axis of symmetry, and determine the domain and range.</p> <ul> <li>a) <code class='latex inline'> \displaystyle f(x) =2(x - 3)^2 - 5 </code></li> <li>b) <code class='latex inline'> \displaystyle f(x) =-3(x - 5)^2 - 1 </code></li> <li>c) <code class='latex inline'> \displaystyle f(x) =-(x + 1)^2 + 6 </code></li> <li>d) <code class='latex inline'> \displaystyle f(x) = (x + 5)^2-3 </code></li> </ul>
<p>Find </p> <ul> <li>Vertex</li> <li>Axis of Symmetry</li> <li>Determine if it opens up/down</li> <li>Range</li> <li>Sketch</li> </ul> <p><code class='latex inline'> \displaystyle f(x) = 4(x + 5)^2 -3 </code></p>
<p>Determine the zeros of each function.</p><p><code class='latex inline'> \displaystyle g(x) = (x + 3)^2 - 15 </code></p>
<p>Describe the transformation(s) that were applied to the graph of <code class='latex inline'>y = x^2</code> to obtain the graph not labelled <code class='latex inline'>y = x^2</code>. Write the equation of the black graph.</p><img src="/qimages/3044" />
<p>Determine the vertex, the axis of symmetry, the direction the parabola opens, and the number of zeros for each quadratic function. Sketch a graph of each.</p><p><code class='latex inline'> \displaystyle f(x) =3(x -5)^2 -2 </code></p>
<p>Sketch the graphs of each of the following: <code class='latex inline'>\displaystyle y=(x-2)^{2} </code></p>
<p>Find </p> <ul> <li>Vertex</li> <li>Axis of Symmetry</li> <li>Determine if it opens up/down</li> <li>Range</li> <li>Sketch</li> </ul> <p><code class='latex inline'> \displaystyle f(x) = -3(x + 5)^2 + 2 </code></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'> y = x^2</code> is translated 2 units to the right. </p>
<p>Sketch each quadratic relation by applying the correct sequence of transformations to the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'>y = 0.5(x + 2)^2 -5</code></p>
<p> Write an equation of the quadratic equation that satisfies each set of conditions.</p><p>The parabola has a minimum value of <code class='latex inline'>4</code> at <code class='latex inline'>x = -2</code>.</p>
<p>Write an equation for the parabola with vertex (6,-2), opening downward, and with no vertical stretch or compression. </p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-x^{2}+4.7 </code></p>
<p>Write the equation of a parabola that matches each description.</p><p>The graph of <code class='latex inline'>y=x^2</code> is stretched vertically by a factor of 5 and then translated 2 units left. 1</p>
<p>Determine the values of <code class='latex inline'>h</code> and <code class='latex inline'>k</code> for each ofthe following transformations. Write the equation in the form <code class='latex inline'>y = (x - b)^2 + k</code>. Sketch the graph.</p><p>The parabola moves 4 units left and 6 units up.</p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y = x^2</code> is translated 5 units upward. </p>
<p>For each quadratic relation in </p> <ul> <li><code class='latex inline'>y=x^2-3</code></li> <li><code class='latex inline'>y=(x+5)^2</code></li> <li><code class='latex inline'>\displaystyle{y=-\frac{1}{2}x^2}</code></li> <li><code class='latex inline'>y=4(x+2)^2-16</code></li> </ul> <p><strong>i)</strong> the direction in which the parabola opens</p><p><strong>ii)</strong> the coordinates of the vertex</p><p><strong>iii)</strong> the equation of the axis of symmetry</p>
<p>Write in standard form the equation of the parabola passing through the given points.</p><p><code class='latex inline'>\displaystyle (-1,-6),(-3,-4),(2,6) </code></p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=x^{2}-2 </code></p>
<p>Graph each parabola. State the coordinates of the vertex. Find any intercepts.</p><p><code class='latex inline'>\displaystyle y=x^{2}-9 </code></p>
<p>Determine the equation of each quadratic relation in vertex form.</p><p>a) vertex at (7, 5), Opens downward, vertical stretch of 4</p><p>b) zeros at 1 and 5, minimum value of - 12, passes through (6, 15)</p>
<p>Write an equation of a parabola that satisfies each set of conditions. </p><p>opens ubward. congruent with <code class='latex inline'>\displaystyle y=2 x^{2} </code>, vertex <code class='latex inline'>\displaystyle (4,-1) </code></p>
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=4 x^{2} </code>; maximum on the <code class='latex inline'>\displaystyle x </code>-axis; axis of symmetry <code class='latex inline'>\displaystyle x=3 </code></p>
<p>Sketch the graph of the quadratic relation. Describe the transformation from the graph of <code class='latex inline'>y = x^2</code>.</p><p><code class='latex inline'>\displaystyle y =- \frac{1}{4}x^2 </code></p>
<p>Without graphing, determine how many <code class='latex inline'>\displaystyle x </code> -intercepts each function has.</p><p><code class='latex inline'>\displaystyle y=-2 x^{2}+3 x-1 </code></p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = (x+5)^2 - 3</code></p>
<ol> <li>Sketch the graph of each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-x^{2}+3 </code></p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=2 x^{2}-3 </code></p>
<p>Write the equation for each of the following quadratic functions.</p><p>Vertex <code class='latex inline'>\displaystyle (-1,2) </code>, opening down, congruent to <code class='latex inline'>\displaystyle y=4 x^{2} </code></p>
<p>Find an equation for the parabola with vertex (2,5) and <em>y</em>-intercept -3. </p>
<p>Sketch the graph of each equation by correctly applying the required transformation(s) to points on the graph of <code class='latex inline'>y = x^2</code>. Use a separate grid for each graph.</p><p><code class='latex inline'> \displaystyle y =-0.25x^2 </code></p>
<p>Describe the graph of <code class='latex inline'> y=a(x-h)^{2}+k </code> if</p><p><code class='latex inline'>\displaystyle k=0 </code></p>
<ol> <li>Communication Describe what happens to the point <code class='latex inline'>\displaystyle (2,4) </code> on the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> when each pair of transformations is applied to the parabola in the given order.</li> </ol> <p>a vertical stretch of scale factor 2, followed by a vertical translation of 5</p>
<p>For each of the following, state the condition on <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> such that the parabola <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> has the given property.</p><p>The parabola does not intersect the <code class='latex inline'>\displaystyle x </code>-axis.</p>
<p>The quadratic relation <code class='latex inline'>y = 2(x + 4)^2 - 7</code> is translated 5 units right and 3 units down. What is the minimum value of the new relation?</p> <ul> <li>Write the equation of this relation in vertex form.</li> </ul>
<p>Sketch the graphs of these four quadratic relations on the same set of axes. </p><p>a) <code class='latex inline'>y = 3x^2</code></p><p>b) <code class='latex inline'>y = \dfrac{1}{3}x^2</code></p><p>c) <code class='latex inline'>y = -2x^2</code></p><p>d) <code class='latex inline'>y = -\dfrac{1}{2}x^2</code></p>
<p>Find the x-intercepts and the vertex of each quadratic relation. Then, sketch its graph.</p><p><code class='latex inline'>\displaystyle y = x^2 -4 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions. </p><p>vertex <code class='latex inline'>\displaystyle (-1,4), y </code>-intercept of 2</p>
<p>Determine the equation of the parabola with vertex <code class='latex inline'>(1, 6)</code> and that passes through <code class='latex inline'>(0, -7)</code></p>
<p>Use a graphing calculator or graphing software to graph each function. State its domain and range. <code class='latex inline'>\displaystyle f(x)=x^{2}-6 </code></p>
<p>Write an equation of a parabola that satisfies each set of conditions.</p><p>opens upward, wider than <code class='latex inline'>\displaystyle y=x^{2} </code>, and vertex <code class='latex inline'>\displaystyle (1,0) </code></p>
<p>Describe the transformations in order that you would apply to the graph of <code class='latex inline'>y = x^2</code> to sketch each quadratic relation.</p><p><code class='latex inline'> \displaystyle y = -3(x - 1)^2 </code></p>
<p>Sketch each parabola and estimate any intercepts.</p><p><code class='latex inline'>\displaystyle y=(x-3)^{2}-1 </code></p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = (x-1)^2 - 8</code></p>
<p>A graphing calculator was used together with the vertex form <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> to graph the screens shown. For the set of graphs on each screen, tell which of the variables <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code> remained constant and which changed. Give possible values for the variables that remained constant.</p><img src="/qimages/163480" />
<p>For each quadratic relation, state</p> <ul> <li>i) the equation of the axis of symmetry</li> <li>ii) the coordinates of the vertex</li> <li>iii) the y-intercept</li> <li>iv) the zeros</li> <li>v) the maximum or minimum value</li> </ul> <p> <img src="/qimages/759" /></p>
<p>Graph and then write in standard form.</p><p><code class='latex inline'> \displaystyle f(x) = 2(x - 3)^2 + 1 </code></p>
<p>Find an equation for the parabola with vertex (-3,-4) that passes through the point (2,6). </p>
<p>The following transformations were applied to the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> : a reflection in the <code class='latex inline'>\displaystyle x </code>-axis, a vertical stretch by a factor of 2 , and a horizontal shift 3 units left. What is the equation of the transformed graph?</p><p><code class='latex inline'>\displaystyle \begin{array}{ll}\text { A. } y=2(x+3)^{2} & \text { C. } y=-2(x-3)^{2} \\ \text { B. } y=-2 x^{2}-3 & \text { D. } y=-2(x+3)^{2}\end{array} </code></p>
<p>The graphs of <code class='latex inline'>\displaystyle f(x)=x^{2} </code> (in green) and another parabola (in black) are shown.</p><p>a) Draw a combination of transformations that would produce the second parabola from the first.</p><p>b) Determine a possible equation for the second parabola.</p><img src="/qimages/160137" /><img src="/qimages/160138" />
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=3 x^{2} ; </code> minimum at <code class='latex inline'>\displaystyle (4,-2) </code></p>
<p>Describe the transformations in order that you would apply to the graph of <code class='latex inline'>y = x^2</code> to sketch each quadratic relation.</p><p><code class='latex inline'> \displaystyle y = 4(x -2)^2 - 5 </code></p>
<p>Write in standard form. </p><p><code class='latex inline'> \displaystyle f(x) = 2-3(x -2)^2 -4 </code></p>
<p>The graph of <code class='latex inline'>\displaystyle y=a x^{2}(a \neq 1, a > 0) </code> is either a vertical stretch or a vertical compression of the graph of <code class='latex inline'>\displaystyle y=x^{2} </code>. Use graphing technolog to determine whether changing the value of <code class='latex inline'>\displaystyle a </code> has a similar effect on the graphs of equations such as <code class='latex inline'>\displaystyle y=a x, y=a x^{3}, y=a x^{4} </code>, and <code class='latex inline'>\displaystyle y=a x^{\frac{1}{2}} </code>.</p>
<ol> <li>The functions <code class='latex inline'>\displaystyle y=m(x-3)^{2}+1 </code> and <code class='latex inline'>\displaystyle y=n(x-2)^{2}-3 </code> are graphed on the same set of axes. How do <code class='latex inline'>\displaystyle m </code> and <code class='latex inline'>\displaystyle n </code> compare if the graphs both open up and a) the graphs are congruent?</li> </ol> <p>b) the first graph is narrower than the second? c) the first graph is wider than the second?</p>
<p>Write one sentence that compares each pair of graphs. </p><p><code class='latex inline'>\displaystyle y=x^{2} </code> and <code class='latex inline'>\displaystyle y=3 x^{2} </code></p>
<p>Copy and complete the following table.</p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|l|l|} \hline \text{Equation } & \text{ Direction of Opening (upward/ downward) } & \text{ Description of Transformation (stretch/ compress)} & \text{ Shape of Graph Compared with Graph of } y=x^{2} (wider/narrower) \\ \hline y=5 x^{2} & & & \\ \hline y=0.25 x^{2} & & & \\ \hline y=-\frac{1}{3} x^{2} & & & \\ \hline y=-8 x^{2} & & & \\ \hline \end{array} </code></p>
<ol> <li>a) Sketch the graphs of each pair of functions. Compare the parabolas in each pair.</li> </ol> <p><code class='latex inline'>\displaystyle y=(x-1)^{2} </code> and <code class='latex inline'>\displaystyle y=(1-x)^{2} </code> <code class='latex inline'>\displaystyle y=(x-4)^{2}-2 </code> and <code class='latex inline'>\displaystyle y=(4-x)^{2}-2 </code> b) Explain your results by expanding <code class='latex inline'>\displaystyle (x-h)^{2} </code> and <code class='latex inline'>\displaystyle (h-x)^{2} </code></p>
<p>A graphing calculator was used together with the vertex form <code class='latex inline'>\displaystyle y=a(x-h)^{2}+k </code> to graph the screens shown. For the set of graphs on each screen, tell which of the variables <code class='latex inline'>\displaystyle a, h </code>, and <code class='latex inline'>\displaystyle k </code> remained constant and which changed. Give possible values for the variables that remained constant.</p><p>M</p>
<p>Find <code class='latex inline'> a </code> and <code class='latex inline'> k </code> so that the given points lie on the parabola.</p><p><code class='latex inline'>\displaystyle y=a(x-1)^{2}+k ;(2,6),(3,12) </code></p>
<p>Describe how the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of the given quadratic functions differ from the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-coordinates of corresponding points of <code class='latex inline'>\displaystyle y=x^{2} </code>.</p><p><code class='latex inline'>\displaystyle y=(x+7)^{2} </code></p>
<p>Determine an equation to represent the parabola.</p><img src="/qimages/8179" />
<p>For the graph, write the quadratic equation in vertex form.</p><img src="/qimages/2080" />
<p>For each quadratic relation,</p> <ol> <li>state the stretch/compression factor and the horizontal/vertical translations</li> <li>determine whether the graph is reflected m‘ the x—axis</li> <li>state the vertex and the equation of the axis of symmetry</li> </ol> <p><code class='latex inline'> \displaystyle y = (x- 2)^2 + 1 </code></p>
<p>Graph each function.</p><p><code class='latex inline'>\displaystyle y=x^{2}-4 </code></p>
<p>Write an equation that defines each parabola.</p><p>congruent to <code class='latex inline'>\displaystyle y=0.4 x^{2} ; </code> opens up; vertex at <code class='latex inline'>\displaystyle (-3,-3) </code></p>
<p>The following transformations are applied to a parabola with the equation <code class='latex inline'>y=2(x+3)^2-1</code> Determine the equation that will result after each transformation.</p> <ul> <li>a translation 4 units right</li> </ul>
<p>Determine any x-intercepts, to the nearest tenth. </p><p><code class='latex inline'>\displaystyle y=-x^{2}+3 </code></p>
<p>Match each equation with its corresponding graph. Explain how you made your decision.</p><p>a) <code class='latex inline'>\displaystyle y=-(x-2)^{2}-3 </code></p><p>c) <code class='latex inline'>\displaystyle y=x^{2}+5 </code></p><p>e) <code class='latex inline'>\displaystyle y=(x-2)^{2} </code></p><p>b) <code class='latex inline'>\displaystyle y=-0.5 x^{2}-4 </code></p><p>d) <code class='latex inline'>\displaystyle y=2(x+2)^{2} </code></p><p>f) <code class='latex inline'>\displaystyle y=-\frac{1}{3}(x+4)^{2}+2 </code></p><img src="/qimages/163384" />
<p>Determine whether each quadratic function intersects the <code class='latex inline'>x</code>—axis at one point, two points, or not at all. Do not draw the graph.</p><p><code class='latex inline'>g(x)=4(x-6)^2+2</code></p>
<ol> <li>Without sketching each parabola, state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-(x+4)^{2} </code></p>
<p>The point <code class='latex inline'>(p, q)</code> lies on the parabola <code class='latex inline'>y = ax^2</code>. If you did not know the value of <code class='latex inline'>a</code>, how could you use the values of <code class='latex inline'>p</code> and <code class='latex inline'>q</code> to determine whether the parabola is wider or narrower than <code class='latex inline'>y = x^2</code>?</p>
<p>Describe the transformations you would apply to the graph of <code class='latex inline'>y=x^2</code>, in the order you would apply them, to obtain the graph of each quadratic relation.</p><p><code class='latex inline'>\displaystyle{y=-\frac{1}{2}x^2}</code></p>
<p>Compare and Contrast Describe the differences between the graphs of <code class='latex inline'>\displaystyle y=(x+6)^{2} </code> and <code class='latex inline'>\displaystyle y=(x-6)^{2}+7 . </code></p>
<p>Find an equation for the parabola with vertex (-6,-2) that passes through the point (-3,-11). </p>
<p>Sketch the graph of <code class='latex inline'>y=x^2+4</code>. Are there any <code class='latex inline'>x</code>—intercepts? Use this information to explain why <code class='latex inline'>a^2+b^2</code> cannot be factored.</p>
<p>Use technology to graph each quadratic relation below. Then determine</p> <ul> <li>i) the equation of the axis of symmetry</li> <li>ii) the coordinates of the vertex</li> <li>iii) the y-intercept</li> <li>iv) the zeros</li> <li>v) the maximum or minimum value</li> </ul> <p><code class='latex inline'>y=-x^2+4</code></p>
<p>For each function, determine the vertex and two points that satisfy the equation. Use the information to sketch the graph of each.</p><p><code class='latex inline'> \displaystyle y = -2(x + 3)^2 + 5 </code></p>
<ol> <li>Sketch each parabola and state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=(x-3)^{2} </code></p>
<p>Graph and then write in standard form.</p><p><code class='latex inline'> \displaystyle f(x) = -(x + 1)^2 - 3 </code></p>
<ol> <li>Without sketching each parabola, state the direction of the opening, the coordinates of the vertex, the equation of the axis of symmetry, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=(x-5)^{2} </code></p>
<ol> <li>Write the equation of the image of <code class='latex inline'>\displaystyle y=3(x-2)^{2}+1 </code> that results from a) a reflection in the <code class='latex inline'>\displaystyle x </code>-axis b) a reflection in the <code class='latex inline'>\displaystyle y </code>-axis c) a reflection in the <code class='latex inline'>\displaystyle y </code>-axis, followed by a reflection in the <code class='latex inline'>\displaystyle x </code>-axis</li> </ol>
<p>Write an equation of a parabola that satisfies each set of conditions.</p><p>opens upward, congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, and vertex <code class='latex inline'>\displaystyle (5,0) </code></p>
<p>For each quadratic relation, state the coordinates of the vertex and the direction of opening. Then, determine how many x-intercepts the relation has.</p><p><code class='latex inline'>y=(x-2)^2+3</code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code>, opens upward, and has vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Now find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is</p><p>translated 2 units to the left</p>
<p>Determine algebraically whether <code class='latex inline'>g(x)</code> is a reflection of <code class='latex inline'>f(x)</code>. Check by graphing. </p><p><code class='latex inline'>f(x) = x^{2} + 2</code><br> <code class='latex inline'>g(x) = -x^{2} + 2</code></p>
<p>Consider a parabola <code class='latex inline'>\displaystyle P </code> that is congruent to <code class='latex inline'>\displaystyle y=x^{2} </code> and with vertex <code class='latex inline'>\displaystyle (2,-4) </code>. Find the equation of a new parabola that results if <code class='latex inline'>\displaystyle P </code> is </p><p>translated 2 units down</p>
<ol> <li>Find <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> so that a parabola <code class='latex inline'>\displaystyle y=a x^{2}+k </code> passes through each pair of points.</li> </ol> <p><code class='latex inline'>\displaystyle (-3,11) </code> and <code class='latex inline'>\displaystyle (4,18) </code></p>
<p>When a graph of <code class='latex inline'>y=x^2</code> is transformed, the point <code class='latex inline'>(3,9)</code> moves to <code class='latex inline'>(8,17)</code>. Describe three sets of transformations that could make this happen. For each set, give the equation of the new parabola.</p>
<p>Graph the function.</p><p><code class='latex inline'>\displaystyle y=-x^{2} </code></p>
<p>Write an equation for the quadratic relation that results from each transformation. </p><p>The graph of <code class='latex inline'>y=x^2</code> is translated 4 units to the left. </p>
<p>Sketch each parabola and estimate any intercepts.</p><p><code class='latex inline'>\displaystyle y=(x-2)^{2} </code></p>
<p>Write one sentence that compares each pair of graphs. </p><p><code class='latex inline'>\displaystyle y=x^{2} </code> and <code class='latex inline'>\displaystyle y=x^{2}-4 </code></p>
<p>Write each function in standard form.</p><p> <code class='latex inline'>\displaystyle f(x)=(x-8)^{2}+4 </code></p>
<p>Determine the number of zeros. Do not use the same method for all four parts.</p><p><code class='latex inline'> \displaystyle f(x) = -3(x - 2)^2 + 4 </code></p>
<p>A parabola is defined by the equation <code class='latex inline'>\displaystyle y = -2(x + 1)^2 + 18 </code>. </p><p>a) Without solving, explain how you can tell how many x-intercepts there are.</p><p>b) Given the current form of the equation, what is the easiest way to find the x-intercepts?</p><p>c) How far apart are the x-intercepts?</p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle 3.5+2.2 x^{2}=y </code></p>
<p>Write an equation for each parabola. </p><img src="/qimages/22259" />
<p> Determine the y-intercept, zeros, equation of the axis of symmetry, and vertex of each quadratic relation. </p><p><code class='latex inline'> y = 2(x + 3)^2</code> </p>
<p>For each quadratic relation, state</p> <ul> <li>i) the equation of the axis of symmetry</li> <li>ii) the coordinates of the vertex</li> <li>iii) the y-intercept</li> <li>iv) the zeros</li> <li>v) the maximum or minimum value</li> </ul> <p> <img src="/qimages/761" /></p>
<p>Write an equation for a parabola with the given vertex and given value of <code class='latex inline'> a </code> .</p><p><code class='latex inline'>\displaystyle (0,0) ; a=-6 </code></p>
<p>Determine the vertex and the direction of opening for each quadratic function. Then state the number of zeros.</p><p><code class='latex inline'> \displaystyle f(x) = -4(x + 3)^2 - 5 </code></p>
<ol> <li>Communication Describe what happens to the point <code class='latex inline'>\displaystyle (2,4) </code> on the graph of <code class='latex inline'>\displaystyle y=x^{2} </code> when each pair of transformations is applied to the parabola in the given order.</li> </ol> <p>a reflection in the x-axis, followed by a vertical compression of scale factor <code class='latex inline'>\frac{1}{2}</code></p>
<p>Sketch each parabola.</p><p><code class='latex inline'>\displaystyle y+\frac{6}{5}=3(x-5)^{2}+\frac{1}{5} </code></p>
<ol> <li>Find <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> so that a parabola <code class='latex inline'>\displaystyle y=a x^{2}+k </code> passes through each pair of points.</li> </ol> <p><code class='latex inline'>\displaystyle (-4,-4) </code> and <code class='latex inline'>\displaystyle (2,2) </code></p>
<p>Copy and complete the table for each parabola. Replace the heading for the second column with the equation for the parabola. </p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline { \text{Property} } & y=a(x-h)^{2}+k \\ \hline \text{Vertex} & \\ \hline \text{Axis of symmetry} & \\ \hline \text{Stretch or compression factor relative to} y=x^{2} & \\ \hline \text{Direction of opening} & \\ \hline \text{Values } x \text{ may take} & \\ \hline \text{Values } y \text{ may take} & \\ \hline \end{array} </code></p><p><code class='latex inline'> y = 3(x+7)^2 - 2</code></p>
<p>For the function, state the vertex and whether the function has a maximum or minimum value. Explain how you decided.</p><p><code class='latex inline'> \displaystyle f(x) =-3(x - 5)^2 - 1 </code></p>
<p>Determine the equation of each parabola.</p><img src="/qimages/164811" />
<p>Determine an equation to represent each parabola.</p><img src="/qimages/5556" />
<p>Write the equation of the quadratic function, first in vertex form and then in standard form.</p><p>vertex (— 4, 8) and passing through (2, - 4 )</p>
<p>Determine the equation of each parabola.</p><img src="/qimages/164809" />
<p>Determine one of the zeros of the quadratic relation <code class='latex inline'>\displaystyle{y=\left(x-\frac{k}{2}\right)^2-\frac{(k-2)^2}{4}}</code></p>
<p>For each graph, state the y-intercept, the zeros, the coordinates of the vertex, and the equation of the axis of symmetry.</p><p> <img src="/qimages/755" /></p>
<ol> <li>Without graphing each function, state the direction of the opening, the coordinates of the vertex, the domain and range, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=-5 x^{2} </code></p>
<p>Write one sentence that compares each pair of graphs. </p><p><code class='latex inline'>\displaystyle y=0.25 x^{2} </code> and <code class='latex inline'>\displaystyle y=-0.25 x^{2} </code></p>
<p>List the sequence of steps required to graph each function.</p><p><code class='latex inline'>\displaystyle f(x)=\frac{1}{3} x^{2}-3 </code></p>
<p>For each quadratic relation,</p> <ol> <li>state the stretch/compression factor and the horizontal/vertical translations</li> <li>determine whether the graph is reflected m‘ the x—axis</li> <li>state the vertex and the equation of the axis of symmetry</li> </ol> <p><code class='latex inline'> \displaystyle y = 2(x+ 1)^2 - 8 </code></p>
<ol> <li>For each parabola, state the direction of the opening, how the parabola is stretched or shrunk, the coordinates of the vertex, the equation of the axis of symmetry, and the maximum or minimum value.</li> </ol> <p><code class='latex inline'>\displaystyle y=2(x-1)^{2} </code></p>
<ol> <li>Communication For quadratic functions of the form <code class='latex inline'>\displaystyle y=a x^{2}+k </code>, where <code class='latex inline'>\displaystyle x </code> and <code class='latex inline'>\displaystyle y </code> are real numbers, describe any relationships between the values of <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle k </code> and</li> </ol> <p>a) the number of <code class='latex inline'>\displaystyle y </code> -intercepts b) the number of <code class='latex inline'>\displaystyle x </code> -intercepts</p>
<p>Write the equations of two different quadratic relations that match each description.</p><p>The graph opens downward and has a narrower opening than the graph of <code class='latex inline'>y = 5x^2</code>.</p>
<p>The vertex of a quadratic relation is (4,-12).</p><p><strong>a)</strong> Write an equation to describe all parabolas with this vertex.</p><p><strong>b)</strong> A parabola with the given vertex passes through point (13,15). Determine the value of <code class='latex inline'>a</code> for this parabola.</p><p><strong>c)</strong> Write the equation of the relation for part b).</p>
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