12. Q12a
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Similar Question 1
<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>
Similar Question 2
<p>A square has side length <code class='latex inline'>3x</code>. One dimension is increased by <code class='latex inline'>2y</code> and the other is decreased by <code class='latex inline'>2y</code>.</p><p>(c) Calculate the area of the rectangle and change in area if <code class='latex inline'>x</code> represents 8 cm and <code class='latex inline'>y</code> represents 5 cm.</p>
Similar Question 3
<p>Write an equation for the parabola.</p><img src="/qimages/768" />
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>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>The cost function in a computer manufacturing plant is <code class='latex inline'>C(x) = 0.28x^2 - 0.7x + 1</code>, where <code class='latex inline'>C(x)</code> is the cost per house in millions of dollars and <code class='latex inline'>x</code> is the number of items produced per hour in thousands. Determine the minimum production cost.</p>
<p>Each dimension of a square playground is increased by 5 m.</p><p>(c) Find a simplified algebraic expression for the increase in area.</p>
<p>For the relation, explain what each coordinate of the vertex represents and what the zeros represent.</p> <ul> <li>a relation that models the height, <code class='latex inline'>h</code>, of a swing above the ground during one swing, <code class='latex inline'>t</code> seconds after the swing begins to move forward</li> </ul>
<p>Use two methods to determine an algebraic expression to represent the area of the figure. Verify that they are equivalent expressions.</p><img src="/qimages/1807" />
<p>All quadratic functions of the form <code class='latex inline'>y = 2x^2 + bx</code> have some similar properties. </p><p>(a) Choose five different values of b and graph each function.</p><p>(b) What are the similar properties?</p><p>(c) Determine the vertex of each parabola.</p><p>(d) Find the relationship between the vertices of these parabolas.</p>
<p>A company&#39;s profit, in thousands of dollars, on sales of computers is modelled by the function <code class='latex inline'>P(x) = -2(x - 3)^2 + 50</code>, where <code class='latex inline'>x</code> is in thousands of computers sold. The company&#39;s profit, in thousands of dollars, on sales of stereo systems is modelled by the function <code class='latex inline'>P(x) = -(x - 2)(x - 7)</code>, where <code class='latex inline'>x</code> is in thousands of stereo systems sold. Calculate the maximum profit the business can earn.</p>
<p>Each function is the demand function of some item, where <code class='latex inline'>x</code> is the number of items sold, in thousands. Determine</p> <ul> <li><p>i) the revenue function </p></li> <li><p>ii) the maximum revenue in thousands of dollars </p></li> </ul> <p><code class='latex inline'> \displaystyle p(x) = -0.6x + 15 </code></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>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>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 square has side length <code class='latex inline'>3x</code>. One dimension is increased by <code class='latex inline'>2y</code> and the other is decreased by <code class='latex inline'>2y</code>.</p>
<p>A farmer has 450 m of fencing to enclose a rectangular area and divide it into two sections as shown.</p><img src="/qimages/15368" /><p><strong>(a)</strong> Write and equation to express the total area enclosed as a function of the width.</p><p><strong>(b)</strong> Determine the domain and range of this area function.</p><p><strong>(c)</strong> Determine the dimensions that give the maximum area.</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>A function models the effectiveness of a TV commercial. After n viewings, the effectiveness, e, is <code class='latex inline'>e = -\frac{1}{90}n^2 + \frac{2}{3}n</code>. </p><p>(a) Determine the range for the effectiveness and the domain of the number of viewings. Explain your answers for the domain and range.</p><p>(b) Use either completing the square or partial factoring to find the vertex. Is it a minimum or a maximum? Explain.</p><p>(c) What conclusions can you make from this function?</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>A high school is planning to build a new playing field surrounded by a running track. The track coach wants two laps around the track to be <code class='latex inline'>1000</code> m. The football coach wants the rectangular infield area to be as large as possible. Can both coaches be satisfied? Explain your answer. </p><img src="/qimages/884" />
<p>A soccer ball is kicked from the ground. After travelling a horizontal distance of <code class='latex inline'>35</code> m, it just passes over a <code class='latex inline'>1.5</code> m tall fence before hitting the ground <code class='latex inline'>37</code> m form where it was kicked.</p><img src="/qimages/574" /><p>Determine the maximum height of the ball.</p>
<p>Each function is the demand function of some item, where <code class='latex inline'>x</code> is the number of items sold, in thousands. Determine</p> <ul> <li><p>i) the revenue function </p></li> <li><p>ii) the maximum revenue in thousands of dollars </p></li> </ul> <p><code class='latex inline'> \displaystyle p(x) = -1.2x + 4.8 </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>A ticket to a school dance is <code class='latex inline'>\$8</code>. Usually, 300 students attend. The dance committee knows that for every <code class='latex inline'>\$1</code> increase in the price of a ticket, <code class='latex inline'>30</code> fewer students attend the dance. What ticket price will maximize the revenue? </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>A ball is thrown vertically upward with an initial velocity of <code class='latex inline'>v</code> metres per second and is affected by gravity, <code class='latex inline'>g</code>. The height, <code class='latex inline'>h</code>. in metres, of the ball after t seconds is given by <code class='latex inline'>h(t) = -\frac{1}{2}gt^2+vt</code>.</p><p>(a) Show that the ball will reach its maximum height at <code class='latex inline'>t = \frac{v}{g}</code></p><p>(b) Show that the maximum height of the ball will be <code class='latex inline'>\frac{v^2}{2g}</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>The profit <code class='latex inline'>P(x)</code> of a cosmetics company, in thousands of dollars, is given by <code class='latex inline'>P(x) = -5x^2 + 400x -2550</code>, where <code class='latex inline'>x</code> is the amount spent on advertising in thousands of dollars.</p><p><strong>(a)</strong> Determine the maximum profit the company can make.</p><p><strong>(b)</strong> Determine the amount spent on advertising that will result in the maximum profit.</p><p><strong>(c)</strong> What amount must be spent on advertising to obtain a profit of at least <code class='latex inline'>\$4 000 000</code>?</p> <p>Each dimension of a square playground is increased by 5 m.</p><p>(b) Find a simplified algebraic expression for the area of the new playground.</p> <p>The predicted flight path of a toy rocket is defined by the relation <code class='latex inline'>h = -2(d -3)(d -15)</code>, where <code class='latex inline'>d</code> is the horizontal distance, in metres, from a safety wall, and h is the height, in metres, above the ground.</p><p><strong>(a)</strong> Sketch a graph of the path of the rocket.</p><p><strong>(b)</strong> How far from the wall is the rocket when it lands on the ground?</p> <p>The revenue, <code class='latex inline'>R</code>, from concert ticket sales at a local venue is calculated as (number of tickets sold) <code class='latex inline'>\times</code> (price of ticket). The current price. of a ticket is <code class='latex inline'>\$20</code>, and the venue typically sells 100 tickets. For each <code class='latex inline'>\$1</code> increase in ticket price, 10 fewer tickets are sold. So, the revenue can be modelled using the equation <code class='latex inline'>R = (100 -10x)(20+ x)</code>, where <code class='latex inline'>x</code> represents the number of <code class='latex inline'>\$1</code> increases.</p> <ul> <li>What price maximizes the revenue? </li> </ul>
<p>For each pair of revenue and cost functions, determine </p> <ul> <li>i) the profit function </li> <li>ii) the value of <code class='latex inline'>x</code> that maximizes profit</li> </ul> <p><code class='latex inline'> \displaystyle R(x) = -2x^2 + 32x, C(x) = 14x + 45 </code></p>
<p>The height, <code class='latex inline'>h(t)</code>, in meters, of the trajectory of a football is given by <code class='latex inline'>h(t) =2+ 28t -4.9t^2</code>, where <code class='latex inline'>t</code> is the time in flight, in seconds. Determine the maximum heigh of the football and the time when the height is reached.</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>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>A rectangular part of a parking lot is to be fenced off to allow some repairs to be done. The workers have fourteen 3-m sections of pre-assembled fencing to use. They want to create the greatest possible area in which to work.</p> <ul> <li>Show why this produces the greatest area using the given fencing sections, but does not create the greatest area that can be enclosed with 42 m of fencing.</li> </ul>
<p>The radius, <code class='latex inline'>r</code>, of a circle has been increased by <code class='latex inline'>k</code>. Both <code class='latex inline'>r</code> and<code class='latex inline'>k</code> are measured in the same units. Write a formula for the area of the new circle. Expand and simplify.</p>
<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>The revenue, <code class='latex inline'>R</code>, from concert ticket sales at a local venue is calculated as (number of tickets sold) <code class='latex inline'>\times</code> (price of ticket). The current price. of a ticket is <code class='latex inline'>\$20</code>, and the venue typically sells 100 tickets. For each <code class='latex inline'>\$1</code> increase in ticket price, <code class='latex inline'>10</code> fewer tickets are sold. So, the revenue can be modelled using the equation <code class='latex inline'>R = (100 -10x)(20+ x)</code>, where <code class='latex inline'>x</code> represents the number of <code class='latex inline'>\$1</code> increases.</p> <ul> <li>Rewrite the equation in the form <code class='latex inline'>R = a(x -4)(x -s)</code></li> </ul> <p>An electronic store sells an average of <code class='latex inline'>60</code> entertainment systems per month at an average of$<code class='latex inline'>800</code> more than the cost price. For every $<code class='latex inline'>20</code> increase in the selling price, the store sells one fewer system. What amount over then cost price will maximize revenue?</p> <p>A soccer ball is kicked from a point <code class='latex inline'>23</code> m to the left of the. halfway line and lands at a point <code class='latex inline'>17</code> in to the right of the halfway line. It reaches a maximum height of <code class='latex inline'>10</code>m during its parabolic flight.</p><p> Determine an equation to represent the path of the soccer ball.</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>Susan has 90 m of fencing to enclose an area in a petting zoo with two dividers to separate three types of young animals. The three pens are to have the same area.</p><img src="/qimages/558" /><p>Determine the domain and the range for the area function.</p> <p>An architect designed the cross-section of a museum using the quadratic relations <code class='latex inline'> \displaystyle \begin{array}{cccc} &y =-\frac{1}{2}(x - 2)(x + 10), &y = -(x +4)(x - 4) \end{array} </code> and <code class='latex inline'>y = -1.5(x -2)(x - 10)</code>, where y is the height, in metres, of the building, and x is the horizontal distance, in metres, from the middle of the building. </p><p><strong>(a)</strong> Sketch a graph of the. cross section of the museum.</p><p><strong>(b)</strong> How wide. is the entire museum?</p><p><strong>(c)</strong> How tall is each section of the museum.</p> <p>The height of a ball thrown from the top of a building can be approximated by the formula <code class='latex inline'>h=-5t^2+15t+20</code>, where <code class='latex inline'>t</code> is the time, in seconds, and <code class='latex inline'>h</code> is the height,in metres.</p><p>Write the formula in factored form. How can you use the factors to find when the ball lands on the ground?</p> <p>Susan has 90 m of fencing to enclose an area in a petting zoo with two dividers to separate three types of young animals. The three pens are to have the same area.</p><img src="/qimages/558" /><p>Express the area function for the three pens in terms of x.</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 baseball is batted at a height of <code class='latex inline'>1</code> m above the ground and reaches a maximum height of <code class='latex inline'>33</code> m at a horizontal distance of <code class='latex inline'>4</code> m.</p><p>a) Determine an equation to model the path of the baseball.</p><p>b) What is the height of the baseball once it has traveled a horizontal distance of 6 m?</p><p>c) At what other horizontal distance is the baseball at the same height as in part b)?</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>A golf ball is hit, and it lands at a point on the same horizontal plane <code class='latex inline'>53</code> m away. The path of the ball took it just over a 9m tall tree that was <code class='latex inline'>8</code>m in front of the golfer.</p><p><strong>(a)</strong> Assume the ball is hit from the origin of a coordinate plane. Find a quartic function that describes the path of the ball.</p><p><strong>(b)</strong> What is the maximum height of the ball?</p><p><strong>(c)</strong> Is possible to move the origin in this situation on and develop another quadratic function to describe the path? If so, find a second quartic function.</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>The height of a rocket above the ground is modelled by the quadratic function <code class='latex inline'>h(t) = - 4t^2 + 32t</code>, where <code class='latex inline'>h(t)</code> is the height in metres <code class='latex inline'>t</code> seconds after the rocket was launched.</p><p>(a) Graph the quadratic function.</p><p>(b) How long will the rocket be in the air? How do you know?</p><p>(c) How high will the rocket be after 3 s?</p><p>(d) What is the maximum height that the rocket will reach?</p> <p>Alf has <code class='latex inline'>24</code> m of fencing to surround a garden, bounded on one side by the wall of his house. What are the dimensions of the largest rectangular garden that he can enclose?</p> <p>Write an equation for the parabola.</p><img src="/qimages/767" /> <p>A sheet of metal that is 30 cm wide and 6 m long is to be used to make a rectangular eavestrough by bending the sheet along the dotted lines. What value of x maximizes the capacity o the eavestrough?</p> <p>For each pair of revenue and cost functions, determine </p> <ul> <li>i) the profit function </li> <li>ii) the value of <code class='latex inline'>x</code> that maximizes profit</li> </ul> <p><code class='latex inline'> \displaystyle R(x) = -3x^2 + 26x, C(x) = 8x + 18 </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 = 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>The revenue, <code class='latex inline'>R</code>, from concert ticket sales at a local venue is calculated as (number of tickets sold) <code class='latex inline'>\times</code> (price of ticket). The current price. of a ticket is <code class='latex inline'>\$20</code>, and the venue typically sells 100 tickets. For each <code class='latex inline'>\$1</code> increase in ticket price, 10 fewer tickets are sold. So, the revenue can be modelled using the equation <code class='latex inline'>R = (100 -10x)(20+ x)</code>, where <code class='latex inline'>x</code> represents the number of <code class='latex inline'>\$1</code> increases.</p> <ul> <li>Sketch and graph. Which of the following can be the graph below?</li> </ul>
<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>The power, <code class='latex inline'>P</code>, in watts, produces by a solar panel is given by the function <code class='latex inline'>P(I) = -5I^2 + 100I</code>, where <code class='latex inline'>I</code> represents the current, in amperes (A).</p><p>(a) What value of the current will maximize the power?</p><p>(b) What is the maximum power?</p>
<p>Each function is the demand function of some item, where <code class='latex inline'>x</code> is the number of items sold, in thousands. Determine</p> <ul> <li><p>i) the revenue function </p></li> <li><p>ii) the maximum revenue in thousands of dollars </p></li> </ul> <p><code class='latex inline'> \displaystyle p(x) = -4x + 12 </code></p>
<p>Each function is the demand function of some item, where <code class='latex inline'>x</code> is the number of items sold, in thousands. Determine</p> <ul> <li><p>i) the revenue function </p></li> <li><p>ii) the maximum revenue in thousands of dollars </p></li> </ul> <p><code class='latex inline'>p(x) = -x + 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 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>What is the height of the ball at a horizontal distance of 20 m?</li> </ul>
<p>Each dimension of a square playground is increased by 5 m.</p><p>(a) Draw a diagram of the situation.</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>Create a quadratic function machine of the form <code class='latex inline'>f(x) = ax^2 + b</code>. Determine the coordinates of the y-intercept and of one other point that is generated b the machine Trade points with a classmate to determine the function.</p>
<p>Jim has a difficult golf shot to make. His ball is 100m from the hole. He wants the ball to land 5 m in front of the hole, so it can roll to the hole. A 20 m tree is between his ball and the hole, 40 m from the hole and 60 m from Jim&#39;s ball. With the base of the tree as the origin, write an algebraic expression to model the height of the ball if it just clears the top of the tree. </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>Write an equation for each parabola.</p><img src="/qimages/766" />
<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>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>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>The height of a ball thrown from the top of a building can be approximated by the formula <code class='latex inline'>h=-5t^2+15t+20</code>, where <code class='latex inline'>t</code> is the time, in seconds, and <code class='latex inline'>h</code> is the height,in metres.</p><p>Write the formula in factored form. Hint: Remove the GCF first.</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 bridge has two towers that support the centre span of cables rise <code class='latex inline'>118</code> m above the. river and are <code class='latex inline'>564</code> m in apart. The cable reaches its lowest point approximately <code class='latex inline'>46</code> m above the river.</p><p><strong>(a)</strong> Sketch a graph to show the curve of the cable if the origin is centred under the lowest point of the cable at the river?s surface</p><p><strong>(b)</strong> Determine an equation to represent the curve of the. cables in the form <code class='latex inline'>y = a(x - r)(x -s)</code>, if possible. If not, explain why.</p>
<p>A soccer ball is kicked from the ground. After travelling a horizontal distance of <code class='latex inline'>35</code> m, it just passes over a <code class='latex inline'>1.5</code> m tall fence before hitting the ground <code class='latex inline'>37</code> m form where it was kicked.</p><img src="/qimages/574" /><p>How far has the ball travelled horizontally to reach the maximum height?</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>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 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 revenue, <code class='latex inline'>R</code>, from concert ticket sales at a local venue is calculated as (number of tickets sold) <code class='latex inline'>\times</code> (price of ticket). The current price. of a ticket is <code class='latex inline'>\$20</code>, and the venue typically sells 100 tickets. For each <code class='latex inline'>\$1</code> increase in ticket price, <code class='latex inline'>10</code> fewer tickets are sold. So, the revenue can be modelled using the equation <code class='latex inline'>R = (100 -10x)(20+ x)</code>, where <code class='latex inline'>x</code> represents the number of <code class='latex inline'>\$1</code> increases.</p> <ul> <li>What does the <code class='latex inline'>R-</code>intercept represent?</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>A ball is kicked into the air and follows a path described by <code class='latex inline'>h(t) = -4.9t^2 + 6t+ 0.6</code>, where <code class='latex inline'>t</code> is the time, in seconds, and <code class='latex inline'>h</code> is the height, in metres, above the ground. Determine the maximum height of the ball, to the nearest tenth of a metre.</p> <p>For each pair of revenue and cost functions, determine </p> <ul> <li>i) the profit function </li> <li>ii) the value of <code class='latex inline'>x</code> that maximizes profit</li> </ul> <p><code class='latex inline'> \displaystyle R(x) = -x^2 + 24x, C(x) = 12x + 28 </code></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> <ul> <li>What do the equation of a circle and the equation of a parabola have in common?</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>The cost, <code class='latex inline'>C</code>, in dollars, of fuel per month for Sam to operate his truck is given by <code class='latex inline'>C(v) = 0.0029v^2 - 0.48v + 142</code>, where <code class='latex inline'>v</code> represents his average driving speed, in kilometres per hour. Find the most efficient speed at which Sam should drive his truck.</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 + 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>For each pair of revenue and cost functions, determine </p> <ul> <li>i) the profit function </li> <li>ii) the value of <code class='latex inline'>x</code> that maximizes profit</li> </ul> <p><code class='latex inline'> \displaystyle R(x) = -2x^2 + 25x, C(x) = 3x + 17 </code></p> <p>The second span of the Bluewater Bridge in Sarnia, Ontario, is supported by two parabolic arches. Each arch is set in concrete foundations that are on opposite sides of the St. Clair River. The feet of the arches are <code class='latex inline'>281 m</code> apart. The top of each arch rises <code class='latex inline'>71 m</code> above the river. Write a function to model the arch.</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>A square has side length <code class='latex inline'>3x</code>. One dimension is increased by <code class='latex inline'>2y</code> and the other is decreased by <code class='latex inline'>2y</code>.</p><p>(a) Find an algebraic expression for the area of the resulting rectangle. Expand.</p><p>(b) Find an algebraic expression for the change in area. Expand.</p><p>(c) Calculate the area of the rectangle and change in area if <code class='latex inline'>x</code> represents 8 cm and <code class='latex inline'>y</code> represents 5 cm.</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> <p>The predicted flight path of a toy rocket is defined by the relation <code class='latex inline'>h = -2(d -3)[d -15)</code>, where d is the horizontal distance, in metres, from a safety wall, and h is the height, in metres, above the ground.</p> <ul> <li>What is the maximum height of the rocket, and how far, horizontally. is it from the wall at that moment?</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 + 2 </code></p> <p>Write an equation for each parabola.</p><img src="/qimages/764" /> <p>The sum of two number is 10. What is th maximum product of these numbers?</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>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" /> <p>A square has side length <code class='latex inline'>3x</code>. One dimension is increased by <code class='latex inline'>2y</code> and the other is decreased by <code class='latex inline'>2y</code>.</p><p>(c) Calculate the area of the rectangle and change in area if <code class='latex inline'>x</code> represents 8 cm and <code class='latex inline'>y</code> represents 5 cm.</p> <p>The revenue, <code class='latex inline'>R</code>, from concert ticket sales at a local venue is calculated as (number of tickets sold) <code class='latex inline'>\times</code> (price of ticket). The current price. of a ticket is <code class='latex inline'>\$20</code>, and the venue typically sells 100 tickets. For each <code class='latex inline'>\$1</code> increase in ticket price, 10 fewer tickets are sold. So, the revenue can be modelled using the equation <code class='latex inline'>R = (100 -10x)(20+ x)</code>, where <code class='latex inline'>x</code> represents the number of <code class='latex inline'>\$1</code> increases.</p> <ul> <li>What does a negative value of <code class='latex inline'>x</code> represent?</li> </ul>
<p>A rock is thrown straight up in the air from an initial height <code class='latex inline'>h_0</code>, in metres, with the initial velocity <code class='latex inline'>v_0</code>, in metres per second. The height in metres above the ground after <code class='latex inline'>t</code> seconds is given by <code class='latex inline'>h(t) = -4.9t^2 + v_{0}t + h_0</code>. Find an expression for the time is takes the rock to reach its maximum height.</p>
<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>The length and width of a rectangle are represented by <code class='latex inline'>x+2</code> and <code class='latex inline'>9-4x</code>. If <code class='latex inline'>x</code> must be an integer, what are the possible values for the area of the rectangle?</p>
<p>A hall charges $<code class='latex inline'>30</code> per person for a sports banquet when <code class='latex inline'>120</code> people attend. For every <code class='latex inline'>10</code> extra people that attend, the hall will decrease the price by$<code class='latex inline'>1.50</code> per person. What number of people will maximize the revenue for the hall?</p>
<p>Last year, a banquet hall charged $<code class='latex inline'>40</code> per person, and <code class='latex inline'>60</code> people attended the hockey banquet dinner. This year, the hall&#39;s manager has said that forever <code class='latex inline'>10</code> extra people that attend the banquet, they will decrease the price by$<code class='latex inline'>1.50</code> per person. What size group would maximize the profit for the hall this year?</p>
<p>The height of a ball thrown vertically upward from a rooftop is modelled by <code class='latex inline'>h(t) = -5t^2 + 20t + 50</code>, where <code class='latex inline'>h(t)</code> is the ball&#39;s height above the ground, in metres, at time <code class='latex inline'>t</code> seconds after the throw.</p><p><strong>(a)</strong> Determine the maximum height of the ball.</p><p><strong>(b)</strong> How long does it take for the ball to reach its maximum height?</p><p><strong>(c)</strong> How high is the rooftop?</p>
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