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<p>The average ticket price at a regular movie theatre (all ages) from 2015 to 2019 can be modelled by <code class='latex inline'>C = 0.06t^2 - 0.27t + 5.36</code>, where <code class='latex inline'>C</code> is the price in dollars and t is the number of years since 2015 ( <code class='latex inline'>t = 0</code> for 2015, <code class='latex inline'>t = 1</code> for 2016, and so on).</p><p><strong>a)</strong> When were ticket prices the lowest during this period?</p><p><strong>b)</strong> What was the average ticket price in 2018?</p><p><strong>c)</strong> What does the model predict the average ticket price will be in 2030?</p><p><strong>d)</strong> Write the equation for the model in vertex form.</p>

<p>Express each equation in standard form and factors form.</p><p><code class='latex inline'>y = -3(x + 3)^2 + 75</code></p>

<p>Show that the value of <code class='latex inline'>3x^2 - 6x + 5</code> cannot be less than 1.</p>

<p>Find the equation of the parabola.</p><img src="/qimages/234" />

<p>A parabola was a <code class='latex inline'>y-</code>intercept of <code class='latex inline'>-4</code> and passes through pints <code class='latex inline'>(-2, 8)</code> and <code class='latex inline'>(1, -1)</code>. Determine the vertex of the parabola. </p>

<p>Graph each function. State the direction of opening, the vertex, and the equation of the axis of symmetry. </p><p><code class='latex inline'>
\displaystyle
\begin{array}{cccccc}
&(a) &f(x) = x^2 - 3 & (c)& f(x) = 2(x - 4)(x + 2) \\
&(b) &f(x) = -(x + 3)^2 - 4 & (d) & f(x) = -\frac{1}{2}x^2 + 4 \\
\end{array}
</code></p>

<p>Write each equation of the parabola below in standard form and factored form.</p><img src="/qimages/1537" />

<p>Use the given information to determine the equation of each quadratic relation in vertex form, <code class='latex inline'>y = a(x - k)^2 + k</code>.</p><p><code class='latex inline'> a = 2, \text{ vertex at } (0, 3)</code></p>

<p>Describe two ways in which the functions <code class='latex inline'>f(x) = 2x^2 - 4x</code> and <code class='latex inline'>g(x) = - (x -1)^2 + 2</code> are alike, and two ways in which they are different. </p>

<p>Find the equation of the parabola.</p><img src="/qimages/232" />

<p>Find the equation of the parabola.</p><img src="/qimages/235" />

<p>Graph each function. State the direction of opening, the vertex, and the equation of the axis of symmetry. </p><p><code class='latex inline'>
\displaystyle
f(x) = 2(x - 4)(x + 2)
</code></p>

<p>The underside of a bridge forms a parabolic arch. The arch has a maximum height of 30 m and a width of 50. Can a sailboat pass under the bridge, 8 m from the axis of symmetry, if the top of its mast is 27 m above the water? Justify your solution.</p>

<p>A parabola has equation <code class='latex inline'>y=(x+2)^2</code></p><p>(c) Verify that the coordinates of the vertex satisfy the equation from part (b).</p><p>(Part b)</p><p>(b) Expand and simplify the equation.</p>

<p>Determine the equation of the parabola in vertex form.</p><img src="/qimages/1535" />

<p>For each quadratic function, state the maximum or minimum value of where it will occur.</p><p><code class='latex inline'>
\displaystyle
f(x) = -3(x - 4)^2 + 7
</code></p>

<p>Use the given information to determine the equation of each quadratic relation in vertex form, <code class='latex inline'>y = a(x - k)^2 + k</code>.</p><p><code class='latex inline'>a= 0.5, \text{ vertex at } (-3.5, 18.3)</code></p>

<p>Determine the equation of a quadratic relation in vertex form, given the following information.</p>
<ul>
<li>vertex at (2, 0), passes through (5, 9)</li>
</ul>

<p>Express each quadratic function in standard form. State the <code class='latex inline'>y</code>-intercept of each.</p><p><code class='latex inline'>
\displaystyle
f(x) = -3(x - 1)^2 + 6
</code></p>

<p>Determine the equation of the parabola in vertex form.</p><img src="/qimages/1536" />

<p>Determine the equation of a quadratic relation in vertex form, given the following information.</p>
<ul>
<li>vertex at <code class='latex inline'>(5, -3)</code>, passes through <code class='latex inline'>(1, -8)</code></li>
</ul>

<p>Consider the quadratic function <code class='latex inline'>f(x) = 4(x - 2)(x + 6)</code>.</p><p>Determine the coordinates of the vertex.</p>

<p>Describe the characteristics that the members of the family of parabolas <code class='latex inline'>f(x) = a(x+ 3)^2 -4</code> have in common. Which member passes through the point (-2, 6)?</p>

<p>Determine the equation of a quadratic relation in vertex form, given the following information.</p>
<ul>
<li>vertex at <code class='latex inline'>(0, 3)</code>, passes through <code class='latex inline'>(2, -5)</code></li>
</ul>

<p>A rectangular swimming pool has a row of water fountains along each of its two longer sides. The two rows of fountains are 10 m apart. Each fountain sprays an identical parabolic—shaped stream of water a total horizontal distance of 8 m toward the opposite Side. Looking from one end of the pool, the streams of water from the two sides cross each other in the middle of the pool at a height of 3 m.</p><p><strong>a)</strong> Determine an equation that represents a stream of water from the left side and another equation that represents a stream of water from the right side. Graph both equations on the same set of axes.</p><p><strong>b)</strong> Determine the maximum height of the water.</p><img src="/qimages/1539" />

<p>Determine the equation of the quadratic relation that corresponds to each graph. Each graph has the same shape as <code class='latex inline'>y = x^2</code>.</p><img src="/qimages/6988" />

<p>A bridge is going to be constructed over a river. The underside of the a bridge will form a parabolic arch, as shown in the picture. The river is 18 m wide and the arch will be anchored on the ground, 3 m back from the riverbank on both sides. The maximum height of the arch must be between 22 m and 26 m above the surface of the river. Create
two different equations to represent arches that satisfy these conditions. Then use graphing technology to graph your equations on the same grid.</p><img src="/qimages/1538" />

<p>Use the given information to determine the equation of each quadratic relation in vertex form, <code class='latex inline'>y = a(x - k)^2 + k</code>.</p><p><code class='latex inline'>a = -3, \text{ vertex at } (2, 0)</code></p>

<p>Examine the parabola at the bottom.</p><p>(a) State the direction of opening.</p><p>(b) Find the coordinates of the vertex.</p><p>(c) List the values of the x-intercepts.</p><p>(d) State the domain and range of the function.</p><p>(e) If you calculated the second differences, what would their sign be? How
do you know?</p><img src="/qimages/879" />

<p>Use the given information to determine the equation of each quadratic relation in vertex form, <code class='latex inline'>y = a(x - k)^2 + k</code>.</p><p><code class='latex inline'>a= -1, \text{ vertex at } (3, -2)</code></p>

<p>Write each equation of the parabola below in standard form and factored form.</p><img src="/qimages/1534" />

<p>The quadratic relation <code class='latex inline'>y = 2(x + 4)^22 - 7</code> is translated 5 units right and 3 units down. What is the minimum value of the new relation?</p>
<ul>
<li>Write the equation of this relation in vertex form.</li>
</ul>

<p> Find the vertex of the following parabolas and state which parabola(with its vertex at the origin) it is congruent to. </p><p><code class='latex inline'>\displaystyle
y = 3x^2 - 24x + 40
</code></p>

<p>Determine the equation of a quadratic relation in vertex form, given the following information.</p>
<ul>
<li>vertex at <code class='latex inline'>(-3, 2)</code>, passes through <code class='latex inline'>(- 1, 14)</code></li>
</ul>

<p>Determine the equation of the parabola in vertex form.</p><img src="/qimages/1534" />

<p>Write each equation of the parabola below in standard form and factored form.</p><img src="/qimages/1535" />

<p>The equation of a parabola is <code class='latex inline'>y = a(x - 1)^2 + q</code>, and the points <code class='latex inline'>(-1, -9)</code> and <code class='latex inline'>(1, 1)</code> lie on the parabola. Determine the maximum value of <code class='latex inline'>y</code>.</p>

<p>Express each equation in standard form and factors form.</p><p><code class='latex inline'>y = 2(x + 1)^2 -18</code></p>

<p>A movie theatre can accommodate a maximum of 350 moviegoers per day. The theatre operators have been changing th admission price to find out how price affects ticket sales and profit. Currently, they charge <code class='latex inline'>\$11</code> a person and sell about 300 tickets per day. After reviewing their data, the theatre operators discovered that they could express the relation between profit, P, and the number of <code class='latex inline'>\$1</code> price increase, <code class='latex inline'>x</code>, as <code class='latex inline'>P = 20(15 - x)(11+ x)</code></p><p><strong>a)</strong> Determine the vertex form of the profit equation.</p><p><strong>b)</strong> What ticket price results in the maximum profit? What is the maximum profit? About how many tickets will be sold at this price?</p>

<p> Find the vertex of the following parabolas and state which parabola(with its vertex at the origin) it is congruent to. </p><p><code class='latex inline'>\displaystyle
y = -5x^2 - 20x - 30
</code></p>

<p>Find the equation of the parabola.</p><img src="/qimages/233" />

<p>Determine the equation of the parabola in vertex form.</p><img src="/qimages/1537" />

<p>Esther claims that the standard form of a quadratic relation is best for solving problems where you need to determine the maximum or minimum value, and that they vertex form is best to use to determine a parabola's zeros. Do you agree or disagree? Explain.</p>

<p>Express each equation in standard form and factors form.</p><p><code class='latex inline'>y = -(x + 1)^2 + 1</code></p>

<p>For each quadratic function, state the maximum or minimum value of where it will occur.</p><p><code class='latex inline'>f(x) = 4x(x + 6)</code></p>

<p>Determine the equation of the quadratic relation that corresponds to each graph. Each graph has the same shape as <code class='latex inline'>y = x^2</code>.</p><img src="/qimages/6989" />

<p>A dance club has a <code class='latex inline'>\$5</code> cover charge and averages 300 customers
on Friday nights. Over the past several months, the club has changed the cover price several times to see how this affects the number
of customers. For every increase of <code class='latex inline'>\$0.50</code> in the cover charge, the number of customers decreases by 30. Use an algebraic model to determine the cover charge that maximizes revenue.</p>

<p>The graph of <code class='latex inline'>y = -2(x + 5)^2 + 8</code> is translated so that its new zeros are -4 and 2. Determine the translation that was applied to the original graph.</p>

<p>Express each equation in standard form and factors form.</p><p><code class='latex inline'>y = (x - 4)^2 -1</code></p>

<p>Examine the parabola at the bottom.</p><p>(a) State the direction of opening.</p><p>(b) Find the coordinates of the vertex.</p><p>(c) What is the equation of the axis of symmetry?</p><p>(d) State the domain and range of the function.</p><p>(e) If you calculated the second differences, what would their sign be? Explain.</p><img src="/qimages/878" />

<p>Write each equation of the parabola below in standard form and factored form.</p><img src="/qimages/1536" />

<p>Determine the maximum or minimum value for each.</p><p><strong>a)</strong> <code class='latex inline'>y = a(x - k)^2 + k</code> with <code class='latex inline'> a = 2, \text{ vertex at } (0, 3)</code>.</p><p><strong>b)</strong> <code class='latex inline'>y = a(x - k)^2 + k</code> with <code class='latex inline'>a = -3, \text{ vertex at } (2, 0)</code>.</p><p><strong>c)</strong> <code class='latex inline'>y = a(x - k)^2 + k</code> with <code class='latex inline'>a= -1, \text{ vertex at } (3, -2)</code>.</p><p><strong>d)</strong> <code class='latex inline'>y = a(x - k)^2 + k</code> with <code class='latex inline'>a= 0.5, \text{ vertex at } (-3.5, 18.3)</code>.</p>

<p>A quadratic relation has zeros at —2 and 8, and a y-intercept of 8.</p><p>Determine the equation of the relation in vertex form.</p>