5. Q5a
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Similar Question 1
<p>Determine whether there are translations and reflections that have equal effects.</p><p><strong>(a)</strong> Graph the function <code class='latex inline'>f(x)=(x-4)^2</code>.</p><p><strong>(b)</strong> Graph the reflection of <code class='latex inline'>f(x)</code> in the <code class='latex inline'>y</code>-axis.</p><p><strong>(c)</strong> Determine a translation that can be applied to <code class='latex inline'>f(x)</code> that has the same effect as the reflection in part (b)</p><p><strong>(d)</strong> Verify algebraically that the transformations in part (b) and (c) are the same.</p>
Similar Question 2
<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 compressed vertically by a factor of <code class='latex inline'>\frac{1}{5}</code>.</p>
Similar Question 3
<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>
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>Compare the graphs of <code class='latex inline'>y = (x - 2)^2</code> and <code class='latex inline'>y = (2 - x)^2</code>. Explain any similarities and differences.</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 to the right.</p>
<p>a) Find an equation for the parabola with vertex <code class='latex inline'>(1, 4)</code> that passes through the point <code class='latex inline'>(3. 8)</code>.</p><p>b) Find an equation for the parabola with vertex <code class='latex inline'>(-2, 5)</code> and <code class='latex inline'>y-</code>intercept <code class='latex inline'>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 y = (x - 2)^2 -4 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>A parabola has equation <code class='latex inline'>y = 2(x - 4)^2-1</code>. Write an equation for the parabola after each set of transformations.</p> <ul> <li>a reflection in the y-axis</li> </ul>
<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 left.</p>
<p>Copy each graph of <code class='latex inline'>f(x)</code> and sketch its reflection in the <code class='latex inline'>x</code>-axis, <code class='latex inline'>g(x)</code>. Then, state the domain and range of each function. </p><img src="/qimages/808" />
<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 y = (x - 4)^2 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </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>Write an equation for each parabola.</p><img src="/qimages/764" />
<p>Sketch the graphs.</p><p><strong>(a)</strong> <code class='latex inline'> \displaystyle y = (x - 4)^2 </code></p><p><strong>(b)</strong> <code class='latex inline'> \displaystyle y = (x - 2)^2 -4 </code></p><p><strong>(c)</strong> <code class='latex inline'> \displaystyle y = (x + 3)^2 -2 </code></p><p><strong>(d)</strong> <code class='latex inline'> \displaystyle y = \frac{1}{2}(x + 1)^2 + 5 </code></p><p><strong>(e)</strong> <code class='latex inline'> \displaystyle y = (x - 7)^2 - 3 </code></p><p><strong>(f)</strong> <code class='latex inline'> \displaystyle y =- (x - 1)^2 + 7 </code></p><p><strong>(g)</strong> <code class='latex inline'> \displaystyle y = 2(x - 4)^2 - 5 </code></p><p><strong>(h)</strong> <code class='latex inline'> \displaystyle y = -3(x + 4)^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 stretched vertically by a factor of <code class='latex inline'>8</code>.</p>
<p>Sketch the graph of each parabola. Label at least three points on 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 = -\frac{1}{2}x^2 </code></p>
<p>Sketch the graph of its reflection in the <code class='latex inline'>y-</code>axis, <code class='latex inline'>h(x)</code>. Then, state the domain and range of each function. </p><img src="/qimages/808" />
<p>Sketch graphs of these three quadratic relations on the same set of axes.</p><p><strong>(a)</strong> <code class='latex inline'> \displaystyle y = (x - 9)^2 </code></p><p><strong>(b)</strong> <code class='latex inline'> \displaystyle y = (x + 2)^2 </code></p><p><strong>(c)</strong> <code class='latex inline'> \displaystyle y = (x - 5)^2 </code></p>
<p>Use words and function notation to describe the transformation that can be applied to the graph of <code class='latex inline'>f(x)</code> to obtain the graph of <code class='latex inline'>g(x)</code>. State the domain and range of each function. </p><img src="/qimages/800" />
<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 y = (x + 3)^2 -2 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>The graph of <code class='latex inline'>f(x)=x^2</code> is transformed to the graph of <code class='latex inline'>g(x)=f(x+8)+12</code>.</p><p><strong>(b)</strong> Determine three points on the base function. Horizontally translate and then vertically translate the points to determine the image points on <code class='latex inline'>g(x)</code>.</p><p><strong>(c)</strong> Start with your original points, but this time reverse the order of your translations. Determine whether the order of the translations is important. </p>
<p> Make tables of values for <code class='latex inline'>y = x^2</code>, <code class='latex inline'>y= 2x^2</code>, <code class='latex inline'>y=x^2 + 1</code>,and <code class='latex inline'>y= (x-3)^2</code>.</p>
<p>Determine whether there are translations and reflections that have equal effects.</p><p><strong>(a)</strong> Graph the function <code class='latex inline'>f(x)=(x-4)^2</code>.</p><p><strong>(b)</strong> Graph the reflection of <code class='latex inline'>f(x)</code> in the <code class='latex inline'>y</code>-axis.</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 y = -3(x + 4)^2 - 2 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>Write an equation for the parabola.</p><img src="/qimages/767" />
<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 y = 2(x - 4)^2 - 5 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>Write an equation for each parabola.</p><img src="/qimages/766" />
<p>A parabola has equation <code class='latex inline'>y = 2(x - 4)^2-1</code>. Write an equation for the parabola after each set of transformations.</p> <ul> <li>a translation of 4 units to the left</li> </ul>
<p>Write an equation for the parabola.</p><img src="/qimages/768" />
<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 y = (x - 7)^2 - 3 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>Sketch the graph of its reflection in the <code class='latex inline'>y-</code>axis, <code class='latex inline'>h(x)</code>. Then, state the domain and range of each function. </p><img src="/qimages/804" />
<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 y =- (x - 1)^2 + 7 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>Sketch graphs of the pair of transformed functions, along with the graph of the parent function, on the same set of axes. </p><p><code class='latex inline'> \displaystyle f(x)=(-\frac{1}{2}x)^2, g(x)=(-\frac{1}{3}x)^2 </code></p>
<p>A parabola has equation <code class='latex inline'>y = 2(x - 4)^2-1</code>. Write an equation for the parabola after each set of transformations.</p> <ul> <li>a reflection in the x-axis.</li> </ul>
<p>Sketch the graph of each parabola. Label at least three points on 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 = 4x^2 </code></p>
<p>The transformations to graph <code class='latex inline'>y = ax^2</code> and <code class='latex inline'>y = x^2 + k</code> both follow what is indicated by the operation, but in <code class='latex inline'>y = (x -h)^2</code>, the transformation is opposite to what the operation seems to indicate.</p><p><strong>(a)</strong> Explain why this might be so.</p><p><strong>(b)</strong> Describe the transformation you would use to graph <code class='latex inline'>y = (2x)^2</code>.</p>
<p>Sketch graphs of the pair of transformed functions, along with the graph of the parent function, on the same set of axes. Describe the transformations in words and note any invariant points.</p><p><code class='latex inline'> \displaystyle y=(-2x)^2, y=(-5x)^2 </code></p>
<p>Sketch the graph of each parabola. Label at least three points on 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 = \frac{2}{3}x^2 </code></p>
<p>Sketch the graph of each parabola. Label at least three points on 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 - 8)^2 </code></p>
<p>A parabola has equation <code class='latex inline'>y = 2(x - 4)^2-1</code>. Write an equation for the parabola after each set of transformations.</p> <ul> <li>a reflection in the x-axis, followed by a translation of 3 units upward</li> </ul>
<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 compressed vertically by a factor of <code class='latex inline'>\frac{1}{5}</code>.</p>
<p>The graph of <code class='latex inline'>y = x^2</code> is reflected in the x-axis, compressed vertically by a factor of <code class='latex inline'>\frac{1}{2}</code>, and then translated 2 units upward. Sketch the parabola and write its equation.</p>
<p>Determine whether there are translations and reflections that have equal effects.</p><p><strong>(a)</strong> Graph the function <code class='latex inline'>f(x)=(x-4)^2</code>.</p><p><strong>(b)</strong> Graph the reflection of <code class='latex inline'>f(x)</code> in the <code class='latex inline'>y</code>-axis.</p><p><strong>(c)</strong> Determine a translation that can be applied to <code class='latex inline'>f(x)</code> that has the same effect as the reflection in part (b)</p><p><strong>(d)</strong> Verify algebraically that the transformations in part (b) and (c) are the same.</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 3 units to the right.</p>
<p>Copy each graph of <code class='latex inline'>f(x)</code> and sketch its reflection in the <code class='latex inline'>x</code>-axis, <code class='latex inline'>g(x)</code>. Then, state the domain and range of each function. </p><img src="/qimages/804" />
<p>Sketch the graph of each parabola. Label at least three points on 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>
<p>Sketch the graph of each parabola. Label at least three points on 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 + 3)^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 downward.</p>
<p>Sketch graphs of the pair of transformed functions, along with the graph of the parent function, on the same set of axes. </p><p><code class='latex inline'> \displaystyle f(x)=(\frac{1}{2}x)^2, g(x)=(\frac{1}{3}x)^2 </code></p>
<p>Sketch each set of functions on the same set of axes.</p><p><code class='latex inline'>y=x^2, y = 3x^2, y = 3(x-2)^2 + 1</code></p>
<p>Write an equation for the parabola with vertex at <code class='latex inline'>(-3, 0)</code>, opening downward, and with a vertical stretch of factor 2.</p>
<p>Sketch the graph of the function <code class='latex inline'>k(x)=-f(-x)</code>. Then, state the domain and range of each function.</p><img src="/qimages/808" />
<p>The graphs of <code class='latex inline'>y=x^2</code> and another parabola are shown.</p><img src="/qimages/866" /><p><strong>(a)</strong> Determine a combination of transformations that would produce the second parabola from the first.</p><p><strong>(b)</strong> Determine a possible equation for the second parabola.</p>
<p>Write an equation for each parabola.</p><img src="/qimages/765" />
<p>Write an equation for the parabola with vertex at <code class='latex inline'>(2, 3)</code>, opening upward, and with no vertical stretch.</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 to the left.</p>
<p>The graph of <code class='latex inline'>f(x)=x^2</code> is transformed to the graph of <code class='latex inline'>g(x)=f(x+8)+12</code>.</p><p>Describe the two transformations represented by this transformation.</p>
<p>Determine algebraically whether <code class='latex inline'>g(x)</code> is a reflection of <code class='latex inline'>f(x)</code> in each case. Verify your answer by graphing. </p><p><code class='latex inline'>f(x)=(x+5)^2+4</code>, <code class='latex inline'>g(x)=-(x+5)^2-4</code></p>
<p>Sketch graphs of these three quadratic relations on the same set of axes.</p><p><strong>(a)</strong> <code class='latex inline'> \displaystyle y = -3x^2 </code></p><p><strong>(b)</strong> <code class='latex inline'> \displaystyle y = \frac{1}{4}x^2 </code></p><p><strong>(c)</strong> <code class='latex inline'> \displaystyle y = -\frac{1}{4}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 y = \frac{1}{2}(x + 1)^2 + 5 </code></p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|} \hline 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 may take} & \\ \hline \text{Values y may take} & \\ \hline \end{array} </code></p>
<p>The graph of <code class='latex inline'>y = x^2</code> is stretched vertically by a factor of 3 and then translated 2 units to the left and 1 unit down. Sketch the parabola and write its equation.</p>
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