12. Q12
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Similar Question 1
<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 - 5 </code></p>
Similar Question 2
<p>Given that <code class='latex inline'>a + b = 21</code> and <code class='latex inline'>\frac{1}{a} + \frac{1}{b} = \frac{7}{18}</code>, find the value of <code class='latex inline'>ab</code>.</p>
Similar Question 3
<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" />
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>If <code class='latex inline'>f(x) = x^2</code>, sketch the graph of each function and state the domain and range.</p><p><code class='latex inline'>y= 0.5f(3 (x - 4)) - 1</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>The path of a soccer ball is modelled by the relation <code class='latex inline'>h = -\frac{1}{16}(d - 28)^2 + 49</code>, where <code class='latex inline'>d</code> is the horizontal distance, in meters, after it was kicked, and <code class='latex inline'>h</code> is the height, in metres, above the ground. </p> <ul> <li>What is the horizontal distance when this occurs?</li> </ul>
<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><img src="/qimages/763" />
<p>If <code class='latex inline'>f(x) = x^2</code>, sketch the graph of each function and state the domain and range.</p><p><code class='latex inline'>y= -f(\frac{1}{4}(x + 1)) + 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. </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>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>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>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>A locus is a set of points that satisfy a specific condition. For example, a circle is a set of points that are equidistant from a fixed point (the centre). Find an equation for the locus of points that is equidistant from the point <code class='latex inline'>(3, 2)</code> and the line <code class='latex inline'>y = -5</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>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><img src="/qimages/763" />
<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>For each function <code class='latex inline'>f(x)</code>, determine the equation for <code class='latex inline'>g(x)</code>. </p><p><code class='latex inline'>f(x)=(x+1)^2-4</code>, <code class='latex inline'>g(x)=f(-x)</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^2 - 5 </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>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>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>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><img src="/qimages/763" />
<p>If <code class='latex inline'>f(x) = (x - 2)(x + 5)</code>, determine the x-intercepts for the function <code class='latex inline'> y=f(-(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>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>The graph shows the path of a rocket fired from the deck of a barge in Lake Ontario at a Canada Day fireworks display. It is a parabola, where h is the height, in metres, of the rocket above the water and t is the time, in seconds.</p><img src="/qimages/772" /> <ul> <li>What is the maximum height reached by the rocket? Justify your answer.</li> </ul>
<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>The grass in the backyard of a house is a square with side length <code class='latex inline'>10</code> m. A square patio is placed in the centre. If the side length, in metres, of the patio is <code class='latex inline'>x</code>, then the area of grass remaining is given by the relation <code class='latex inline'>A = -x^2 + 100</code>.</p><img src="/qimages/1814" /> <ul> <li>Find the intercepts. What do they represent?</li> </ul>
<p>The path of a soccer ball is modelled by the relation <code class='latex inline'>h = -\frac{1}{16}(d - 28)^2 + 49</code>, where <code class='latex inline'>d</code> is the horizontal distance, in meters, after it was kicked, and <code class='latex inline'>h</code> is the height, in metres, above the ground. </p> <ul> <li>Find the other time when the vertical distance is equal top when the horizontal distance was 20m.</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 3 units to the right.</p>
<p>A stadium roof has a cross section in the shape of a parabolic arch with equation <code class='latex inline'>y = -\frac{1}{45}x^2 + 20</code>. Which graph represents the arch? Justify your reasoning.</p><img src="/qimages/2498" />
<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>Write an equation for each parabola.</p><img src="/qimages/764" />
<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p> <ul> <li></li> </ul>
<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>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>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>The path of a soccer ball is modelled by the relation <code class='latex inline'>h = -\frac{1}{16}(d - 28)^2 + 49</code>, where <code class='latex inline'>d</code> is the horizontal distance, in meters, after it was kicked, and <code class='latex inline'>h</code> is the height, in metres, above the ground. </p> <ul> <li>What is the maximum height of the ball?</li> </ul>
<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>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^2</code>, <code class='latex inline'>g(x)=(-x)^2</code></p>
<p>Use the base function <code class='latex inline'>f(x)=x^2</code>. Write the equation for each transformed function.</p><p><strong>(a)</strong> <code class='latex inline'>n(x)=f(x-4)-6</code></p><p><strong>(b)</strong> <code class='latex inline'>r(x)=f(x+2)+9</code></p><p><strong>(c)</strong> <code class='latex inline'>s(x)=f(x+6)-7</code></p><p><strong>(d)</strong> <code class='latex inline'>t(x)=f(x-11)+4</code></p>
<p> What is the equation of each circle?</p> <ul> <li>radius <code class='latex inline'>5</code>, centered at <code class='latex inline'>(0, 3)</code></li> <li>radius <code class='latex inline'>7</code>, centred at <code class='latex inline'>(6, 1)</code></li> <li>radius <code class='latex inline'>8</code>, centred at <code class='latex inline'>(-3, 5)</code></li> <li>radius <code class='latex inline'>r</code>, centred at <code class='latex inline'>(h, k)</code></li> </ul>
<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>The cables of a suspension bridge form parabolas. If the minimum point of the centre cable is placed at the origin, determine an equation for each parabola. Describe the values of x for which each equation is valid.</p><img src="/qimages/771" />
<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>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>For each graph, describe the reflection that transforms <code class='latex inline'>f(x)</code> into <code class='latex inline'>g(x)</code>. </p><img src="/qimages/382" />
<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><img src="/qimages/763" />
<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>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>If <code class='latex inline'>f(x) = x^2</code>, sketch the graph of each function and state the domain and range.</p><p><code class='latex inline'>y=f(x - 2) + 3</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>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p> <ul> <li></li> </ul>
<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><img src="/qimages/763" />
<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><img src="/qimages/763" />
<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^2 + 8 </code> </p><p><strong>(b)</strong> <code class='latex inline'> \displaystyle y = x^2 - 5 </code> </p><p><strong>(c)</strong> <code class='latex inline'> \displaystyle y = x^2 - 10 </code> </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>The path of a soccer ball is modelled by the relation <code class='latex inline'>h = -\frac{1}{16}(d - 28)^2 + 49</code>, where <code class='latex inline'>d</code> is the horizontal distance, in meters, after it was kicked, and <code class='latex inline'>h</code> is the height, in metres, above the ground. </p> <ul> <li>Sketch the path of the soccer ball.</li> </ul>
<p>If <code class='latex inline'>f(x) = (x - 2)(x + 5)</code>, determine the x-intercepts for the function <code class='latex inline'>y=f(-\frac{1}{3}x)</code>.</p>
<p>For each function <code class='latex inline'>f(x)</code>, determine the equation for <code class='latex inline'>g(x)</code>. </p><p><code class='latex inline'>f(x)=(x-5)^2+9</code>, <code class='latex inline'>g(x)=-f(-x)</code></p>
<p>Parabolic mirrors are used in telescopes to help magnify the image. The cross section of a parabolic mirror is shown.</p><img src="/qimages/770" /><p><strong>(a)</strong> Sketch a graph to represent the cross section of the mirror, placing the vertex at (0, -0.24).</p><p><strong>(b)</strong> Write an equation to represent the cross section of the mirror. Describe the values of x for which your equation is valid.</p><p><strong>(c)</strong> Move the mirror to a different location on the same set of axes. Write a different equation to represent the cross section of the mirror. Describe the values of x for which your equation is valid.</p>
<p>Write an equation for each parabola.</p><img src="/qimages/766" />
<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>Given that <code class='latex inline'>a + b = 21</code> and <code class='latex inline'>\frac{1}{a} + \frac{1}{b} = \frac{7}{18}</code>, find the value of <code class='latex inline'>ab</code>.</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/804" />
<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>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>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>The flight path of a firework is modelled by the relation <code class='latex inline'>h = -5(t -5)^2 + 127</code> where <code class='latex inline'>h</code> is the height, in metres, of the firework above the ground and <code class='latex inline'>t</code> is the time, in seconds, since the firework was fired.</p><p><strong>(a)</strong> What is the maximum height reached by the firework? How many seconds after it was fired does the firework reach this height?</p><p><strong>(b)</strong> How high was the firework above the ground when it was fired?</p>
<p>For <code class='latex inline'>f(x) = x^2</code>, sketch the graph of <code class='latex inline'>g(x) = f(2x + 6)</code>.</p>
<ul> <li>What do the equation of a circle and the equation of a parabola have in common?</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 = x^2 + 0.5 </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 y = \frac{1}{2}(x + 1)^2 + 5 </code></p><img src="/qimages/763" />
<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>
<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p><p><em>(a)</em> What is the length of the skid mark for a car travelling at 50 km/h? 100 km/h?</p><p><strong>answer b) only</strong></p><p><strong>(b)</strong> How do the results in part a) compare?</p>
<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><img src="/qimages/763" />
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