Now You Try

<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>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 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 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>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>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>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>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> 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>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>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> 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>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 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>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>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 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>

<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" />