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<em><strong>
<p>Students will complete the following questions to practice the skills they have learned in this lesson.</p>
</strong></em>
<ol class="os-raise-noindent">
<li>Here are graphs of functions \(f\) and \(g\).</li>
</ol>
<p>Each represents the height of an object being launched into the air as a function of time.</p>
<p><img alt="Graphs of functions f and g. Vertical axis, height. Horizontal axis, time. Function f begins low on the y axis, increases in height quickly, then ends at about halfway down the x axis. Function g begins high on the y axis, moves gradually downward, and ends toward the end of the x axis." height="184" src="https://k12.openstax.org/contents/raise/resources/c2fb42c0f8f727ba961e42c44dcb9437c595eeeb" width="284"></p>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Which object was launched from a higher point?
<ul>
<li> The object represented by function \(f\) </li>
<li> The object represented by function \(g\) </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The object represented by function \(g\) because its \(y\)-intercept is above the \(y\)-intercept of the graph of \(f\).</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Which object reached a higher point?
<ul>
<li> The object represented by function \(f\) </li>
<li> The object represented by function \(g\) </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The object represented by function \(f\) because the vertex is higher.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> Which object was launched with the greater upward velocity?
<ul>
<li> The object represented by function \(f\) </li>
<li> The object represented by function \(g\) </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The object represented by function \(f\) because the “slope” near the \(y\)-intercept is greater. It increases in height much more quickly than the other object.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> Which object landed last?
<ul>
<li> The object represented by function \(f\) </li>
<li> The object represented by function \(g\) </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The object represented by function \(g\) because the positive \(x\)-intercept of its graph is to the right of the positive \(x\)-intercept of the graph of \(f\).</p>
</ol>
<ol class="os-raise-noindent" start="2">
<li>The function \(h\) given by \(h(t)=(1−t)(8+16t)\) models the height of a ball in feet, \(t\) seconds after it was thrown.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Find one of the zeros of the functions using \((8+16t)\). </li>
</ol><br>
<p><strong>Answer:</strong> -0.5. From the factored form, we can tell that the graph intersects the horizontal axis at \(t=-0.5\). The zero of the function is the same value.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Find one of the zeros of the functions using \((1-t)\). </li>
</ol><br>
<p><strong>Answer:</strong> 1. From the factored form, we can tell that the graph intersects the horizontal axis at \((1-t)\). The zero of the function is the same value.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What do the zeros tell us in this situation?
<ul>
<li> The positive zero means the time, in seconds, when the basketball is on the ground or hits the ground. </li>
<li> The negative zero represents the height, in feet, when the time is zero. </li>
<li> The positive zero represents the time, in seconds, when the basketball reaches the maximum height. </li>
<li> The negative zero represents the time, in seconds, when the basketball is on the ground or hits the ground. </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The positive zero means the time, in seconds, when the basketball is on the ground or hits the ground.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> Are both zeros meaningful?
<ul>
<li> Yes </li>
<li> No </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> No. Only the positive zero is meaningful. It tells us the ball hits the ground after 1 second.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="5" type="a">
<li> From what height, in feet, is the ball thrown? </li>
</ol><br>
<p><strong>Answer:</strong> 8 feet. When \((1−t)(8+16t)\)is rewritten in standard form it is \(8+8t−16t^2\). The constant term (which corresponds to the \(y\)-intercept) tells us the height of the basketball when \(t=0\) or when it is launched.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="6" type="a">
<li> When, in seconds, does the ball reach its highest point? </li>
</ol><br>
<p><strong>Answer:</strong> 0.25 seconds. Graphing \(h\) shows the maximum point at \((0.25,9)\). Or, you could recognize that 0.25 is the \(x\)-coordinate of the vertex because it is halfway between the two zeros, -0.5 and 1.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="7" type="a">
<li> How high, in feet, does the ball go? </li>
</ol><br>
<p><strong>Answer:</strong> 9 feet. Graphing \(h\) shows the maximum point at \((0.25,9)\). Or, you could recognize that 0.25 is the \(x\)-coordinate of the vertex because it is halfway between the two zeros, -0.5 and 1. Then evaluate the function at 0.25 to get a height of 9.</p>
</ol>
<ol class="os-raise-noindent" start="3">
<li>The height in feet of a thrown football is modeled by the equation \(f(t)=6+30t−16t^2\), where time \(t\) is measured in seconds.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Which term in the equation describes how many feet above the ground the football was thrown?
<ul>
<li> The constant term 6 </li>
<li> The linear term \(30t\) </li>
<li> The squared term \(-16t^2\) </li>
<li> The squared term \(16t^2\) </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The constant term 6</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Which term in the equation describes the initial upward velocity of the football in feet per second?
<ul>
<li> The constant term 6 </li>
<li> The linear term \(30t\) </li>
<li> The squared term \(-16t^2\) </li>
<li> The squared term \(16t^2\) </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The linear term \(30t\).</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> How does the squared term \(-16t^2\) affect the value of the function \(f\)? </li>
</ol><br>
<p><strong>Answer:</strong> The squared term decreases the value of the function because the values of \(16t^2\) are being subtracted from \(6+30t\)</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="4" type="a">
<li> What does this term \(−16t^2\) reveal about the situation?
<ul>
<li> The influence of gravity pulling the ball down to the ground. </li>
<li> The initial upward velocity of the football. </li>
<li> How far the football was thrown above the ground. </li>
<li> How long the ball is in the air before it hits the ground. </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> The influence of gravity pulling the ball down to the ground.</p>
</ol>
<ol class="os-raise-noindent" start="4">
<li>The height in feet of an arrow is modeled by the equation \(h(t)=(1+2t)(18−8t)\), where \(t\) is seconds after the arrow is shot.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> At what time, in seconds, does the arrow hit the ground? </li>
</ol><br>
<p><strong>Answer: </strong>2.25. The positive zero is 2.25 because \((18−8t)\) is equal to zero when \(t=2.25\). The zeros mean the times, in seconds, when the arrow is on the ground. Only the positive zero makes sense. The negative zero would represent a time before the arrow was launched.</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> At what height, in feet, is the arrow shot? </li>
</ol><br>
<p><strong>Answer:</strong> 18 feet. When the arrow is shot, \(t\) is 0 and \(h(0)=(1+2⋅0)(18−8⋅0)=(1)(18)=18\).</p>
</ol>
<p>5. Two objects are launched into the air.</p>
<p>The height, in feet, of Object A is given by the equation \(f(t)=4+32t−16t^2\).</p>
<p>The height, in feet, of Object B is given by the equation \(g(t)=2.5+40t−16^2\).</p>
<p>In both functions, \(t\) is seconds after launch.</p>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Which object was launched from a greater height?
<ul>
<li> Object A </li>
<li> Object B </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> Object A. The objects are launched when \(t=0\). \(f(0)=4+32⋅0−16(0)2=4\) and \(g(0)=2.5+40⋅0−16(0)2=2.5\).</p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> Which object was launched with a greater upward velocity?
<ul>
<li> Object A </li>
<li> Object B </li>
</ul>
</li>
</ol><br>
<p><strong>Answer:</strong> Object B. The coefficient of \(t\) gives the upward velocity and 40 is greater than 32.</p>
</ol>