chap6-rev-problems.xml

<?xml version="1.0"?>


<!-- This file was originally part of the book     -->
<!--   Modeling, Functions, and Graphs   -->
<!--           4th                                     -->
<!-- Copyright (C) Katherine Yoshiwara      -->

<exercises xml:id="chap6-rev-problems"  xmlns:xi="http://www.w3.org/2001/XInclude">
    <title>Review Problems</title>

<exercisegroup cols="2"><introduction><p>
    Solve by factoring.
</p></introduction>

<exercise number="1">
    <statement><p><m>x^2 + x = 4 - (x + 2)^2</m></p></statement>
    <answer><p><m>0, ~\dfrac{-5}{2} </m></p></answer>
</exercise>

<exercise number="2"><statement><p><m>(n - 3)(n + 2) = 6 </m></p></statement></exercise>

<exercise number="3"><statement><p><m>x(3x + 2) = (x + 2)^2 </m></p></statement>
<answer><p><m>-1, ~2</m></p></answer>
</exercise>

<exercise number="4"><statement><p><m> 6y = (y + 1)^2 + 3 </m></p></statement></exercise>

<exercise number="5"><statement><p><m>4x - (x + 1) (x + 2) = -8 </m></p></statement>
<answer><p><m>-2, ~3</m></p></answer>
</exercise>

<exercise number="6"><statement><p><m>3(x + 2)^2 = 15 + 12x</m></p></statement></exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>Write a quadratic equation with integer coefficients and with the given solutions.</p></introduction>

<exercise number="7"><statement><p><m>\dfrac{-3}{4} </m> and <m>8</m></p></statement>
<answer><p><m>4x^2 - 29x - 24 = 0</m></p></answer>
</exercise>

<exercise number="8"><statement><p><m>\dfrac{5}{3} </m> and <m>\dfrac{5}{3} </m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup><introduction><p>Graph each equation using the <term>ZDecimal</term> setting. Locate the <m>x</m>-intercepts and use them to write the quadratic expression in factored form.</p></introduction>

<exercise number="9"><statement><p><m>y = x^2 - 0.6x - 7.2</m></p></statement>
<answer><p><m>y= (x - 3) (x + 2.4)</m></p></answer>
</exercise>

<exercise number="10"><statement><p><m>y = -x^2  + 0.7x + 2.6</m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>Use a substitution to solve.</p></introduction>

<exercise number="11"><statement><p><m>2^{2p} - 6 · 2^p + 8 = 0</m></p></statement>
<answer><p><m>1</m>, <m>~2</m></p></answer>
</exercise>

<exercise number="12"><statement><p><m>3^{2r} - 6 · 3^r + 5 = 0</m></p></statement>
</exercise>

<exercise number="13"><statement><p><m>\left(\dfrac{1}{b} \right)^2-3\left(\dfrac{1}{b} \right)-4=0 </m></p></statement>
<answer><p><m>-1</m>, <m>~\dfrac{1}{4} </m></p></answer>
</exercise>


<exercise number="14"><statement><p><m>\left(\dfrac{1}{q} \right)^2+ \dfrac{1}{q} -2=0 </m></p></statement>
</exercise>
</exercisegroup>




<exercisegroup cols="2"><introduction><p>Solve by completing the square.</p></introduction>

<exercise number="15"><statement><p><m>x^2 - 4x - 6 = 0</m></p></statement>
<answer><p><m>2\pm \sqrt{10} </m></p></answer>
</exercise>

<exercise number="16"><statement><p><m>x^2  + 3x = 3</m></p></statement>
</exercise>

<exercise number="17"><statement><p><m>2x^2 + 3 = 6x</m></p></statement>
<answer><p><m>\dfrac{3\pm\sqrt{3} }{2} </m></p></answer>
</exercise>

<exercise number="18"><statement><p><m>3x^2 = 2x + 3</m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>Solve by using the quadratic formula.</p></introduction>

<exercise number="19"><statement><p><m>\dfrac{1}{2}x^2+1=\dfrac{3}{2}x </m></p></statement>
<answer><p><m>1</m>, <m>~2</m></p></answer>
</exercise>

<exercise number="20"><statement><p><m>x^2 - 3x + 1 = 0 </m></p></statement>
</exercise>

<exercise number="21"><statement><p><m>x^2 - 4x + 2 = 0 </m></p></statement>
<answer><p><m>2\pm \sqrt{2} </m></p></answer>
</exercise>

<exercise number="22"><statement><p><m>2x^2 + 2x = 3 </m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>Solve each formula for the indicated variable.</p></introduction>

<exercise number="23"><statement><p><m>K=\dfrac{1}{2}mv^2 </m>, for <m>v</m></p></statement>
<answer><p><m>\pm\sqrt{\dfrac{2K}{m}} </m></p></answer>
</exercise>

<exercise number="24"><statement><p><m>a^2+b^2=c^2 </m>, for <m>b</m></p></statement>
</exercise>

<exercise number="25"><statement><p><m>h = 6t - 3t^2 </m>, for <m>t</m></p></statement>
<answer><p><m>\dfrac{3\pm\sqrt{9-3h} }{3} </m></p></answer>
</exercise>

<exercise number="26"><statement><p><m>D=\dfrac{n^2-3n}{2} </m>, for <m>n</m></p></statement>
</exercise>
</exercisegroup>



<exercise number="27"><statement><p>
    In a tennis tournament among <m>n</m> competitors, <m>\dfrac{n(n-1)}{2} </m> matches must be played. If the organizers can schedule <m>36</m> matches, how many players should they invite?
</p></statement>
<answer><p><m>9</m></p></answer>
</exercise>

<exercise number="28"><statement><p>
    The formula <m>S=\dfrac{n(n+1)}{2} </m> gives the sum of the first <m>n</m> positive integers. How many consecutive integers must be added to make a sum of <m>91</m>?
</p></statement>
</exercise>

<exercise number="29"><statement><p>
    Irene wants to enclose <m>2</m> adjacent chicken coops of equal size against the henhouse wall. She has <m>66</m> feet of chicken wire fencing and would like the total area of the two coops to be <m>360</m> square feet. What should the dimensions of the chicken coops be?
</p></statement>
<answer><p><m>10</m> ft by <m>18</m> ft or <m>12</m> ft by <m>15</m> ft</p></answer>
</exercise>

<exercise number="30"><statement><p>
    The base of an isosceles triangle is one inch shorter than the equal sides, and the altitude of the triangle is <m>2</m> inches shorter than the equal sides. What is the length of the equal sides?
</p></statement>
</exercise>

<exercise number="31"><statement><p>
    A car traveling at <m>50</m> feet per second (about <m>34</m> miles per hour) can stop in <m>2.5</m> seconds after applying the brakes hard. The distance the car travels, in feet, <m>t</m> seconds after applying the brakes is <m>d = 50t - 10t^2</m>. How long does it take the car to travel <m>40</m> feet?
</p></statement>
<answer><p><m>1</m> sec</p></answer>
</exercise>

<exercise number="32"><statement><p>
    You have <m>300</m> feet of wire fence to mark off a rectangular Christmas tree lot with a center divider, using a brick wall as one side of the lot. If you would like to enclose a total area of <m>7500</m> square feet, what should be the dimensions of the lot?
</p></statement>
</exercise>

<exercise number="33"><statement><p>
    The height, <m>h</m>, of an object <m>t</m> seconds after being thrown from ground level is given by
    <me>h=v_0 t -\dfrac{1}{2}gt^2 </me>
    where <m>v_0</m> is its starting velocity and <m>g</m> is a constant that depends on gravity. On the Moon, the value of <m>g</m> is approximately <m>5.6</m>. Suppose you hit a golf ball on the Moon with an upward velocity of <m>100</m> feet per second.
    <ol>
        <li><p>Write an equation for the height of the golf ball <m>t</m> seconds after you hit it.</p></li>
        <li><p>Graph your equation in the window
            <md>
                <mrow>
                    \small{\text{Xmin}} \amp = 0 \amp\amp \small{\text{Xmax}} = 47
                </mrow>
                <mrow>
                    \small{\text{Ymin}} \amp = 0 \amp\amp \small{\text{Ymax}} = 1000
                </mrow>
            </md> 
        </p></li>
        <li><p>Use the <term>Trace</term> to estimate the maximum height the golf ball reaches.</p></li>
        <li><p>Use your equation to calculate when the golf ball will reach a height of <m>880</m> feet.</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>h = 100t - 2.8t^2</m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-33.jpg"  width="80%"><description>parabola</description></image> </p></li>
    <li><p><m>893</m> ft</p></li>
    <li><p><m>15\frac{5}{7}</m> sec on the way up and <m>20</m> sec on the way down</p></li>
</ol></p></answer>
</exercise>

<exercise number="34"><statement><p>
    An acrobat is catapulted into the air from a springboard at ground level. Her height, <m>h</m>, in meters is given by the formula <me>h = -4.9t^2 + 14.7t</me>, where <m>t</m> is the time in seconds from launch. Use your calculator to graph the acrobat’s height versus time. Use the window
    <md>
        <mrow>
            \small{\text{Xmin}} \amp = 0 \amp\amp \small{\text{Xmax}} = 4.7
        </mrow>
        <mrow>
            \small{\text{Ymin}} \amp = 0 \amp\amp \small{\text{Ymax}} = 12
        </mrow>
    </md> 
    <ol>
        <li><p>Use the <term>Trace</term> to find the coordinates of the highest point on the graph. When does the acrobat reach her maximum height, and what is that height?</p></li>
        <li><p>Use the formula to find the height of the acrobat after <m>2.4</m> seconds.
        </p></li>
        <li><p>Use the <term>Trace</term> to verify your answer to part (b). Find another time when the acrobat is at the same height.</p></li>
        <li><p>Use the formula to find two times when the acrobat is at a height of <m>6.125</m> meters. Verify your answers on the graph.</p></li>
        <li><p>What are the coordinates of the horizontal intercepts of your graph? What do these points have to do with the acrobat?</p></li>
    </ol>
</p></statement>
</exercise>



<exercisegroup cols="2"><introduction><p>Show that the shaded areas are equal.</p></introduction>

<exercise number="35"><statement><p><image source="images/fig-chap6-rev-35"  width="75%"><description>two shaded areas</description></image> </p></statement>
<answer><p><m>A_1</m> is the area of a square minus the area of two triangles:
<me>x^2-2\left(\dfrac{1}{2}y\cdot y \right)=x^2-y^2 </me>
</p></answer>
</exercise>

<exercise number="36"><statement><p><image source="images/fig-chap6-rev-36"  width="50%"><description>two shaded areas</description></image> </p></statement></exercise>
</exercisegroup>



<exercisegroup cols="3"><introduction><p><ol label="a">
    <li><p>Find the coordinates of the vertex and the intercepts.</p></li>
    <li><p>Sketch the graph.</p></li>
</ol></p></introduction>

<exercise number="37"><statement><p><m>y=\dfrac{1}{2}x^2 </m></p></statement>
<answer><p><ol>
    <li><p>Vertex and intercepts are all <m>(0,0) </m>.</p></li>
    <li><p><image source="images/fig-ans-chap6-rev-37"  width="40%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="38"><statement><p><m>y = x^2 - 4</m></p></statement></exercise>

<exercise number="39"><statement><p><m>y= x^2 - 9x </m></p></statement>
<answer><p><ol>
    <li><p>Vertex <m>(\frac{9}{2},\frac{-81}{4})</m>; <m>x</m>-intercepts <m>(9,0)</m> and <m>(0,0)</m>; <m>y</m>-intercept <m>(0,0) </m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-39"  width="40%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="40"><statement><p><m>y = -2x^2 - 4x</m></p></statement></exercise>


<exercise number="41"><statement><p><m>y= x^2 + x - 6 </m></p></statement>
<answer><p><ol>
    <li><p>Vertex <m>(\frac{-1}{2},\frac{-25}{4})</m>; <m>x</m>-intercepts <m>(-3,0)</m> and <m>(2,0)</m>; <m>y</m>-intercept <m>(0,-6) </m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-41"  width="40%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="42"><statement><p><m>y = x^2 - 3x + 4</m></p></statement></exercise>

<exercise number="43"><statement><p><m>y = 8 - x - 2x^2 </m></p></statement>
<answer><p><ol>
    <li><p>Vertex <m>(\frac{-1}{4},\frac{65}{8})</m>; <m>x</m>-intercepts <m>\left(\dfrac{-1 \pm \sqrt{65}}{4} ,0 \right)</m>; <m>y</m>-intercept <m>(0,8) </m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-43"  width="40%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="44"><statement><p><m>y = -2x^2 + x - 4</m></p></statement></exercise>

<exercise number="45"><statement><p><m>y = x^2 - x - 9 </m></p></statement>
<answer><p><ol>
    <li><p>Vertex <m>(\frac{1}{2},\frac{-37}{4})</m>; <m>x</m>-intercepts <m>\left(\dfrac{1 \pm \sqrt{37}}{2} ,0 \right)</m>; <m>y</m>-intercept <m>(0,-9) </m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-45"  width="40%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="46"><statement><p><m>y = -x^2 + 2x + 4</m></p></statement></exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>Use the discriminant to determine how many <m>x</m>-intercepts the graph has.</p></introduction>

<exercise number="47"><statement><p><m>y = -2x^2 + 5x - 1</m></p></statement>
<answer><p>Two</p></answer>
</exercise>

<exercise number="48"><statement><p><m>y = -12 - 3x + 4x^2</m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>Use the discriminant to determine the nature of the solution of each equation.</p></introduction>

<exercise number="49"><statement><p><m>4x^2 - 12x + 9 = 0</m></p></statement>
<answer><p>One rational solution</p></answer>
</exercise>

<exercise number="50"><statement><p><m>2t^2 + 6t + 5 = 0</m></p></statement>
</exercise>

<exercise number="51"><statement><p><m>2y^2 = 3y - 4</m></p></statement>
<answer><p>No real solutions</p></answer>
</exercise>

<exercise number="52"><statement><p><m>\dfrac{x^2}{4}=x+\dfrac{5}{4} </m></p></statement>
</exercise>
</exercisegroup>



<exercise number="53"><statement><p>
    The total profit Kiyoshi makes from producing and selling <m>x</m> floral arrangements is
    <me>P(x) = -0.4x^2 + 36x - 400</me>
    <ol>
        <li><p>How many floral arrangements should Kiyoshi produce and sell to maximize his profit? What is his maximum profit?</p></li>
        <li><p>Verify your answers on a graph.</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>45</m>; <m>\$410</m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-53.jpg"  width="80%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="54"><statement><p>
    Lightning does about one billion dollars damage annually in the United States and kills 85 people. To study lightning, meteorologists fire small rockets at passing thunderclouds to induce lightning bolts. The rocket trails a thin copper wire that is vaporized by the lightning, leaving a plasma channel that carries the current to the grounding point. The rocket boosts the wire to a height of <m>250</m> meters, and <m>t</m> seconds later, its height is given in meters by <me>h(t) = -4.9t^2 + 32t + 250</me>.
    <ol>
        <li><p>When does the rocket reach its maximum height? What is the maximum height?</p></li>
        <li><p>Verify your answers on a graph.</p></li>
    </ol>
</p></statement>
</exercise>

<exercise number="55"><statement><p>
    A beekeeper has beehives distributed over <m>60</m> square miles of pastureland. When she places <m>4</m> hives per square mile, each hive produces about <m>32</m> pints of honey per year. For each additional hive per square mile, honey production drops by <m>4</m> pints per hive.
    <ol>
        <li><p>Write an equation for the total production of honey, in pints, in terms of the number of additional hives per square mile.</p></li>
        <li><p>How many additional hives per square mile should the beekeeper install in order to maximize honey production?</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>y = 60 (4 + x) (32 - 4x)</m></p></li>
    <li><p><m>2</m> </p></li>
</ol></p></answer>
</exercise>

<exercise number="56"><statement><p>
    A small company manufactures radios. When it charges <m>\$20</m> for a radio, it sells <m>500</m> radios per month. For each dollar the price is increased, <m>10</m> fewer radios are sold per month.
    <ol>
        <li><p>Write an equation for the monthly revenue in terms of the price increase over <m>\$20</m>.</p></li>
        <li><p>What should the company charge for a radio in order to maximize its monthly revenue?</p></li>
    </ol>
</p></statement>
</exercise>



<exercisegroup cols="2"><introduction><p><ol label="a">
    For Problems 57<ndash/>60,
    <li><p>Find all values of <m>x</m> for which <m>f(x) = 0</m>.</p></li>
    <li><p>Find all values of <m>x</m> for which <m>g(x) = 0</m>.</p></li>
    <li><p>Find all values of <m>x</m> for which <m>f(x) = g(x) </m>.</p></li>
    <li><p>Graph each pair of functions in the same window, then sketch the graph on paper. Illustrate your answers to (a)–(c) as points on the graph.</p></li>
</ol></p></introduction>

<exercise number="57"><statement><p><m>f (x) = 2x^2 + 3x</m>, <m>~g(x) = 5 - 6x</m></p></statement>
<answer><p><ol cols="2">
    <li><p> <m>0, ~\dfrac{-3}{2} </m></p></li>
    <li><p><m>\dfrac{5}{6} </m></p></li>
    <li><p><m>-5, ~\dfrac{1}{2} </m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-57" at="parabola and line" width="80%"/> </p></li>
</ol></p></answer>
</exercise>

<exercise number="58"><statement><p><m>f (x) =3x^2 - 6x</m>, <m>~g(x) = 8 + 4x</m></p></statement></exercise>

<exercise number="59"><statement><p><m>f (x) = 2x^2 -2x</m>, <m>~g(x) = x^2+3</m></p></statement>
<answer><p><ol cols="2">
    <li><p> <m>0, ~1 </m></p></li>
    <li><p>None</p></li>
    <li><p><m>-1, ~3 </m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-59" at="two parabolas" width="80%"/> </p></li>
</ol></p></answer>
</exercise>

<exercise number="60"><statement><p><m>f (x) =x^2 + 4x + 6</m>, <m>~g(x) = 4-x^2</m></p></statement></exercise>
</exercisegroup>



<exercisegroup cols="3"><introduction><p>Solve each inequality algebraically, and give your answers in interval notation. Verify your solutions by graphing.</p></introduction>

<exercise number="61"><statement><p><m>(x - 3) (x + 2)\gt 0</m></p></statement>
<answer><p><m>(-∞,-2) \cup (3,∞)</m>
<image source="images/fig-ans-chap6-rev-61"  width="40%">
  <description>
    quadratic inequality
  </description>
</image> </p></answer>
</exercise>

<exercise number="62"><statement><p><m>y^2 - y - 12\le 0</m></p></statement></exercise>

<exercise number="63"><statement><p><m>2y^2 - y\le 3</m></p></statement>
<answer><p><m>\left[-1,\dfrac{3}{2} \right] </m>
<image source="images/fig-ans-chap6-rev-63"  width="40%">
  <description>
    quadratic inequality
  </description>
</image> </p></answer>
</exercise>

<exercise number="64"><statement><p><m>3z^2 - 5z\gt 2 </m></p></statement></exercise>

<exercise number="65"><statement><p><m>s^2\le 4 </m></p></statement>
<answer><p><m>[-2,2] </m>
<image source="images/fig-ans-chap6-rev-65"  width="40%">
  <description>
    quadratic inequality
  </description>
</image> </p></answer>
</exercise>

<exercise number="66"><statement><p><m>4t^2\gt 12 </m></p></statement></exercise>
</exercisegroup>



<exercise number="67"><statement><p>
    The Sub Station sells <m>220 - \dfrac{1}{4}p</m> submarine sandwiches at lunchtime if it sells them at <m>p</m> cents each.
    <ol>
        <li><p>Write a formula for the Sub Station's daily revenue in terms of <m>p</m>.</p></li>
        <li><p>What range of prices can the Sub Station charge if it wants to keep its daily revenue from subs over <m>\$480</m>? (Remember to convert <m>\$480</m> to cents.)</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>R=p \left(220-\dfrac{1}{4}p \right) </m></p></li>
    <li><p>Between <m>\$4.00</m> and <m>\$4.80</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="68"><statement><p>
    When it charges <m>p</m> dollars for an electric screwdriver, Handy Hardware will sell <m>30 - \dfrac{1}{2}p</m> screwdrivers per month.
    <ol>
        <li><p>Write a formula in terms of <m>p</m> for Handy Hardware's monthly revenue from electric screwdrivers.</p></li>
        <li><p>How much should Handy charge per screwdriver if it wants the monthly revenue from the screwdrivers to be over <m>\$400</m>?</p></li>
    </ol>
</p></statement>
</exercise>



<exercisegroup cols="2"><introduction><p>Solve each system algebraically, and verify your solution with a graph.</p></introduction>

<exercise number="69"><statement><p>
    <m>\begin{aligned}
        y+x^2 \amp = 4 \\
        y \amp = 3
    \end{aligned}</m>
</p></statement>
<answer><p><m>(1, 3)</m>, <m>~(-1, 3)</m>
<image source="images/fig-ans-chap6-rev-69"  width="40%">
  <description>
    parabola and horizontal line
  </description>
</image> </p></answer>
</exercise>

<exercise number="70"><statement><p>
    <m>\begin{aligned}
        y \amp = 3-x^2 \\
        5x+y \amp = 7
    \end{aligned}</m>
</p></statement>
</exercise>

<exercise number="71"><statement><p>
    <m>\begin{aligned}
        y \amp = x^2-5 \\
        y \amp = 4x
    \end{aligned}</m>
</p></statement>
<answer><p><m>(-1, -4)</m>, <m>~(5, 20)</m>
<image source="images/fig-ans-chap6-rev-71"  width="40%">
  <description>
    parabola and line
  </description>
</image> </p></answer>
</exercise>

<exercise number="72"><statement><p>
    <m>\begin{aligned}
        y \amp = x^2 -2x+1 \\
        y \amp = 3-x
    \end{aligned}</m>
</p></statement>
</exercise>

<exercise number="73"><statement><p>
    <m>\begin{aligned}
        y \amp = x^2-6x+20 \\
        y \amp = 2x^2-2x-25
    \end{aligned}</m>
</p></statement>
<answer><p><m>(-9, 155)</m>, <m>~(5, 15)</m>
<image source="images/fig-ans-chap6-rev-73"  width="40%">
  <description>
    two parabolas
  </description>
</image> </p></answer>
</exercise>

<exercise number="74"><statement><p>
    <m>\begin{aligned}
        y \amp = x^2 -5x-28 \\
        y \amp = -x^2+4x+28
    \end{aligned}</m>
</p></statement>
</exercise>

<exercise number="75"><statement><p>
    <m>\begin{aligned}
        y \amp = \frac{1}{2} x^2-\frac{3}{2}x \\
        y \amp = -\frac{1}{2}x^2 +\frac{1}{2}x+3
    \end{aligned}</m>
</p></statement>
<answer><p><m>(-1, 2)</m>, <m>~(3, 0)</m>
<image source="images/fig-ans-chap6-rev-75"  width="40%">
  <description>
    two parabolas
  </description>
</image> </p></answer>
</exercise>

<exercise number="76"><statement><p>
    <m>\begin{aligned}
        y \amp = 2x^2 +5x-3 \\
        y \amp = x^2+4x-1
    \end{aligned}</m>
</p></statement>
</exercise>
</exercisegroup>



<exercise number="77"><statement><p>Find values of <m>a</m>, <m>b</m>, and <m>c</m> so that the graph of the parabola <m>y = ax^2 + bx + c</m> contains the points <m>(-1,-4)</m>, <m>(0, -6)</m>, and <m>(4, 6)</m>.</p></statement>
<answer><p><m>a=1</m>, <m>~b=-1</m>, <m>~c=-6</m></p></answer>
</exercise>

<exercise number="78"><statement><p><ol>
    <li><p>Find values of <m>a</m>, <m>b</m>, and <m>c</m> so that the graph of the parabola <m>y = ax^2 + bx + c</m> contains the points <m>(0,-2)</m>, <m>(-6,1)</m>, and <m>(4, 6)</m>.</p></li>
    <li><p>Plot the data points and sketch the graph on the grid.</p></li>
</ol></p></statement>
</exercise>

<exercise number="79"><statement><p>
    Find a parabola that fits the following data points.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell><m>x</m></cell>
            <cell><m>-8</m></cell>
            <cell><m>-4</m></cell>
            <cell><m>2</m></cell>
            <cell><m>4</m></cell>
        </row>
        <row>
            <cell><m>y</m></cell>
            <cell><m>10</m></cell>
            <cell><m>18</m></cell>
            <cell><m>0</m></cell>
            <cell><m>-14</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
<answer><p><m>p(x)=\dfrac{-1}{2}x^2-4x+10 </m></p></answer>
</exercise>

<exercise number="80"><statement><p>
    Find a parabola that fits the following data points.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell><m>x</m></cell>
            <cell><m>-3</m></cell>
            <cell><m>0</m></cell>
            <cell><m>2</m></cell>
            <cell><m>4</m></cell>
        </row>
        <row>
            <cell><m>y</m></cell>
            <cell><m>-46</m></cell>
            <cell><m>8</m></cell>
            <cell><m>-6</m></cell>
            <cell><m>-60</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
</exercise>

<exercise number="81"><statement><p>Find the equation for a parabola that has a vertex of <m>(15,-6)</m> and passes through the point <m>(3, 22.8)</m>.</p></statement>
<answer><p><m>y = 0.2 (x - 15)^2 -6</m></p></answer>
</exercise>

<exercise number="82"><statement><p>Find the equation for a parabola that has a vertex of <m>(-3,-8)</m> and passes through the point <m>(6, 12.25)</m>.</p></statement>
</exercise>



<exercisegroup cols="2"><introduction><p>
    For Problems 83<ndash/>86,
    <ol label="a">
        <li><p>Write the equation in vertex form.</p></li>
        <li><p>Use transformations to sketch the graph.</p></li>
    </ol>
</p></introduction>

<exercise number="83"><statement><p><m>f (x) = x^2 - 24x + 44</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>f (x) = (x - 12)^2 -100</m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-83"  width="90%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="84"><statement><p><m>g(x) = x^2 + 30x + 300</m></p></statement>
</exercise>

<exercise number="85"><statement><p><m>y = \dfrac{1}{3}x^2 + 2x + 1</m></p></statement>
<answer><p><ol cols="2">
    <li><p><m>y =\dfrac{1}{3} (x +3)^2 -2</m></p></li>
    <li><p><image source="images/fig-ans-chap6-rev-85"  width="90%"><description>parabola</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="86"><statement><p><m>y = -2x^2 + 4x + 3</m></p></statement>
</exercise>
</exercisegroup>



<exercise number="87"><statement><p>
    The height of a cannonball was observed at <m>0.2</m>-second intervals after the cannon was fired, and the data were recorded in the table.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center"><col halign="left"/>
        <row>
            <cell>Time (seconds)</cell>
            <cell><m>0.2</m></cell>
            <cell><m>0.4</m></cell>
            <cell><m>0.6</m></cell>
            <cell><m>0.8</m></cell>
            <cell><m>1.0</m></cell>
            <cell><m>1.2</m></cell>
            <cell><m>1.4</m></cell>
            <cell><m>1.6</m></cell>
            <cell><m>1.8</m></cell>
            <cell><m>2.0</m></cell>
        </row>
        <row>
            <cell>Height (meters)</cell>
            <cell><m>10.2</m></cell>
            <cell><m>19.2</m></cell>
            <cell><m>27.8</m></cell>
            <cell><m>35.9</m></cell>
            <cell><m>43.7</m></cell>
            <cell><m>51.1</m></cell>
            <cell><m>58.1</m></cell>
            <cell><m>64.7</m></cell>
            <cell><m>71.0</m></cell>
            <cell><m>76.8</m></cell>
        </row>
    </tabular></sidebyside>
    <ol>
        <li><p>Find the equation of the least-squares regression line for height in terms of time.</p></li>
        <li><p>Use the linear regression equation to predict the height of the cannonball at <m>3</m> seconds and at <m>4</m> seconds after it was fired.</p></li>
        <li><p>Make a scatterplot of the data and draw the regression line on the same axes.</p></li>
        <li><p>Find the quadratic regression equation for height in terms of time.</p></li>
        <li><p>Use the quadratic regression equation to predict the height of the cannonball at <m>3</m> seconds and at <m>4</m> seconds after it was fired.</p></li>
        <li><p>Draw the quadratic regression curve on the graph from part (c).</p></li>
        <li><p>Which model is more appropriate for the height of the cannonball, linear or quadratic? Why?</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>h = 36.98t + 5.17</m></p></li>
    <li><p><m>116.1</m> m, <m>~153.1</m> m</p></li>
    <li><p><image source="images/fig-ans-chap6-rev-87.jpg"  width="70%"><description>scatterplot and regression line</description></image> </p></li>
    <li><p><m>h= -4.858t^2 + 47.67t + 0.89</m></p></li>
    <li><p><m>100.2</m> m, <m>~113.9</m> m</p></li>
    <li><p><image source="images/fig-ans-chap6-rev-87f.jpg"  width="70%"><description>scatterplot and regression parabola</description></image></p></li>
    <li><p>Quadratic: Gravity will slow the cannonball, giving the graph a concave down shape.</p></li>
</ol></p></answer>
</exercise>

<exercise number="88"><statement><p>
    Max took a sequence of photographs of an explosion spaced at equal time intervals. From the photographs, he was able to estimate the height and vertical velocity of some debris from the explosion, as shown in the table. (Negative velocities indicate that the debris is falling back to Earth.)</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center"><col halign="left"/>
        <row>
            <cell><line>Velocity</line><line>(meters/second)</line></cell>
            <cell><m>67</m></cell>
            <cell><m>47</m></cell>
            <cell><m>27</m></cell>
            <cell><m>8</m></cell>
            <cell><m>-12</m></cell>
            <cell><m>-31</m></cell>
        </row>
        <row>
            <cell><line>Height</line><line>(meters)</line></cell>
            <cell><m>8</m></cell>
            <cell><m>122</m></cell>
            <cell><m>196</m></cell>
            <cell><m>232</m></cell>
            <cell><m>228</m></cell>
            <cell><m>185</m></cell>
        </row>
    </tabular></sidebyside>
    <ol>
        <li><p>Enter the data into your calculator and create a scatterplot. Fit a quadratic regression equation to the data, then graph the equation on the scatterplot.</p></li>
        <li><p>Use your regression equation to find the vertex of the parabola. What do the coordinates represent, in terms of the problem? What should the velocity of the debris be at the maximum height of the debris?</p></li>
    </ol>
</p></statement></exercise>

</exercises>