Now You Try

<p>The table shows the data for a bouncing ball.</p><img src="/qimages/753" /><p><strong>i)</strong> Describe the relation.</p><p><strong>ii)</strong> Draw a curve of best fit.</p>

<p>Use finite differences to determine whether each relation is linear, quadratic, or neither.</p><img src="/qimages/599" />

<p>A rectangle has a width of x centimetres, and its length is double its width.</p><p><strong>i)</strong> Draw a curve of best fit.</p><p><strong>ii)</strong> Explain why the graph of this relation is non-linear.</p>

<p>Which scatter plot(s) could be modelled using a curve instead of a line of best fit? Explain. </p><img src="/qimages/22236" />

<p>For the table, calculate the second differences. Determine whether the function is quadratic.</p><img src="/qimages/6032" />

<p>Does the scatter plot(s) could be modeled using a curve instead of a line of best fit?
Explain.</p><img src="/qimages/748" />

<p>Each table of values defines a parabola. Determine the equation of the axis of symmetry of the parabola, and write the equation in vertex form.</p><img src="/qimages/1548" />

<p>This table shows the number of imported cars that were sold in Newfoundland between 2003 and <code class='latex inline'>\displaystyle 2007 . </code></p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|c|c|c|c|}
\hline Year & 2003 & 2004 & 2005 & 2006 & 2007 \\
\hline Sales of Imported Cars (number sold) & 3996 & 3906 & 3762 & 3788 & 4151 \\
\hline
\end{array}
</code></p><p>a) Create a scatter plot, and draw a quadratic curve of good fit.</p><p>b) Determine an algebraic equation in vertex form to model this relation.</p><p>c) Use your model to predict how many imported cars were sold in 2008 .</p><p>d) What does your model predict for 2006 ? Is this prediction accurate? Explain.</p><p>e) Check the accuracy of your model using quadratic regression.</p>

<p>The data in the table below represent the height of a golf ball at different times.</p><p><strong>a)</strong> Create a scatter plot, and draw a curve of a good fit</p><p><strong>b)</strong> Use your graph for part a) to approximate the zeros of the relation. </p><p><strong>c)</strong> Determine an equation that models this situation.</p><p><strong>d)</strong> Use your equation for part c) to estimate the maximum height of the ball.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline Time (s) & Height (\mathbf{m}) \\
\hline 0.0 & 0.000 \\
\hline 0.5 & 10.175 \\
\hline 1.0 & 17.900 \\
\hline 1.5 & 23.175 \\
\hline 2.0 & 26.000 \\
\hline 2.5 & 26.375 \\
\hline 3.0 & 24.300 \\
\hline 3.5 & 19.775 \\
\hline 4.0 & 12.800 \\
\hline 4.5 & 3.375 \\
\hline
\end{array}
</code></p>

<p>A city opened a new landfill site in 2000. The table shows how much garbage was added to the landfill in each year from 2000 to 2007.</p><img src="/qimages/22215" /><p>a) Determine the total mass of garbage in the landfill at the end of each year.</p><p>b) Make a scatter plot of the total mass of garbage versus the year. Draw a curve of best fit.</p><p>c) What problems do you predict if growth continues at its current rate?</p>

<p>The table shows the data for a bouncing ball.</p><img src="/qimages/753" />
<ul>
<li>How would the relationship change for a ball that was bouncier?</li>
</ul>

<p>Does the scatter plot(s) could be modelled using a curve instead of a line of best fit?
Explain.</p><img src="/qimages/749" />

<p>For the table, calculate the second differences. Determine whether the function is quadratic.</p><img src="/qimages/6031" />

<p>An altimeter is attached to a model rocket before it is launched. The table shows the recorded data from the rocket’s flight.</p><img src="/qimages/751" />
<ul>
<li>Draw a curve of best fit.</li>
</ul>

<p>An altimeter is attached to a model rocket before it is launched. The table shows the recorded data from the rocket’s flight.</p><img src="/qimages/751" />
<ul>
<li>Use your model to predict the height of the rocket after 8s.</li>
</ul>

<p> Use the graph at the right to determine an equation for curve
of good fit. Write the equation in factored and standard forms.</p><img src="/qimages/777" />

<p>A school custodian finds a tennis ball on the roof of the school and throws it to the ground below. The table gives the height of the ball above the ground as it moves through the air.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|c|c|c|c|c|c|}
\hline Time (s) & 0.0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\
\hline Height (\mathbf{m}) & 5.00 & 11.25 & 15.00 & 16.25 & 15.00 & 11.25 & 5.00 \\
\hline
\end{array}
</code></p><p>a) Do the data appear to be linear or quadratic? Explain.</p><p>b) Create a scatter plot, and draw a quadratic curve of good fit.</p><p>c) Estimate the coordinates of the vertex.</p><p>d) Determine an algebraic relation in vertex form to model the data.</p><p>e) Use your model to predict the height of the ball at <code class='latex inline'>\displaystyle 2.75 \mathrm{~s} </code> and <code class='latex inline'>\displaystyle 1.25 \mathrm{~s} </code>.</p><p>f) How effective is your model for time values that are greater than <code class='latex inline'>\displaystyle 3.5 </code> s? Explain.</p><p>g) Check the accuracy of your model using quadratic regression.</p>

<p>The table shows the average fuel economy of a car at a test track.</p><img src="/qimages/752" /><p>a) Make a scatter plot of the data.</p><p>b) Describe the relation.</p><p>c) Draw a curve of best fit.</p><p>d) Use your model to predict the fuel economy at 200 km/h.</p><p>d) This car does not get very good fuel economy. How would a graph of a car with better fuel economy look? Why?</p>

<p>A toy rocket that is sitting on a tower is launched vertically upward. The table shows the height, <code class='latex inline'>\displaystyle h </code>, of the rocket in centimetres at <code class='latex inline'>\displaystyle t </code> seconds after its launch.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|r|r|r|r|r|r|r|r|}
\hline \boldsymbol{t}(\mathbf{s}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline \boldsymbol{h}(\mathrm{cm}) & 88 & 107 & 116 & 115 & 104 & 83 & 52 & 11 \\
\hline
\end{array}
</code></p><p>a) Using a graphing calculator, create a scatter plot to display the data.</p><p>b) Estimate the vertex of your model. Then write the equation of the model in vertex form and standard form.</p><p>c) Use the regression feature on the graphing calculator to create a quadratic model for the data. Compare this model with the model you created for part b).</p><p>d) What is the maximum height of the rocket? When does the rocket reach this maximum height?</p><p>e) When will the rocket hit the ground?</p>

<p>An altimeter is attached to a model rocket before it is launched. The table shows the recorded data from the rocket’s flight.</p><img src="/qimages/751" />
<ul>
<li>Make a scatter plot of the data.</li>
</ul>

<p>An altimeter is attached to a model rocket before it is launched. The table shows the recorded data from the rocket’s flight.</p><img src="/qimages/751" />
<ul>
<li>Describe the relation.</li>
</ul>

<p>Use finite differences to determine whether each relation is linear, quadratic, or neither.</p><img src="/qimages/16994" />

<p>The table shows the data for a bouncing ball.</p><img src="/qimages/753" />
<ul>
<li>Make a scatter plot of the data.</li>
</ul>

<p>Use difference tables to determine whether the data represent a linear or quadratic relationship.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|l|r|r|r|r|r|}
\hline Time (s) & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline Height (\mathrm{m}) & 0 & 15 & 20 & 20 & 15 & 0 \\
\hline
\end{array}
</code></p>

<p><strong>a)</strong> Use the graph at the right to determine an equation for curve
of good fit. Write the equation in factored and standard forms.</p><p><strong>b)</strong> Use your equation to estimate the value of <code class='latex inline'>y</code> when <code class='latex inline'>x=1</code>.</p><img src="/qimages/777" />

<p>The table shows the path of a ball, where x is the horizontal distance, in metres, and h is the height, in metres, above the ground.</p><img src="/qimages/598" /><p>(a) Sketch a graph of the quadratic relation.</p><p>(b) Describe the flight path of the ball. Identify the axis of symmetry and the
vertex.</p><p>(c) What is the maximum height that the ball reached?</p><p>(d) Verify that <code class='latex inline'>h = -x^2 + 8x + 1</code> can be used to model the light path of the ball.</p>

<p>Data for DVD sales in Canada, over several years, are given in the table.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|c|c|c|c|}
\hline Year & 2002 & 2003 & 2004 & 2005 & 2006 \\
\hline x , Years Since 2002 & 0 & 1 & 2 & 3 & 4 \\
\hline DVDs Sold (1000s) & 1446 & 3697 & 4573 & 4228 & 3702 \\
\hline
\end{array}
</code></p><p>a) Using graphing technology, create a scatter plot to display the data.</p><p>b) Estimate the vertex of the graph you created for part a). Then determine an equation in vertex form to model the data.</p><p>c) How many DVDs would you expect to be sold in 2010 ?</p><p>d) Check the accuracy of your model using quadratic regression.</p>

<p>The data in the table show the average mass of a boy as he grows between the ages of 1 and 12 . State the following:</p><p>a) domain</p><p>b) range</p><p>c) whether the relation is a function</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|c|c|c|c|c|}
\hline Age (years) & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline Mass (kg) & 11.5 & 13.7 & 16.0 & 20.5 & 23.0 & 23.0 \\
\hline
\end{array}
</code></p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|c|c|c|c|c|}
\hline Age (years) & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline Mass (kg) & 30.0 & 33.0 & 39.0 & 38.5 & 41.0 & 49.5 \\
\hline
\end{array}
</code></p>

<p>A rectangle has a width of x centimetres, and its length is double its width.</p>
<ul>
<li>Make a scatter plot of the data.</li>
</ul>

<p>Each table of values defines a parabola. Determine the equation of the axis of symmetry of the parabola, and write the equation in vertex form.</p><img src="/qimages/1547" />

<p>The population of a bacteria colony is measured every hour and results in the data shown in the table at the left. Use difference tables to determine whether the number of bacteria, <code class='latex inline'>\displaystyle n(t) </code>, is a linear or quadratic function of time. Explain.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|r|}
\hline Time (h) & Bacteria Count \\
\hline 0 & 12 \\
\hline 1 & 23 \\
\hline 2 & 50 \\
\hline 3 & 100 \\
\hline
\end{array}
</code></p>

<p>State whether each line or curve of best fit is a good model for the data. Justify your answer.</p><img src="/qimages/21743" />

<p>A chain of ice cream stores sells <code class='latex inline'>\displaystyle \$ 840 </code> of ice cream cones per day. Each ice cream cone costs <code class='latex inline'>\displaystyle \$ 3.50 </code>. Market research shows the following trend in revenue as the price of an ice cream cone is reduced.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|l|c|c|c|c|c|c|c|}
\hline Price (\$) & 3.50 & 3.00 & 2.50 & 2.00 & 1.50 & 1.00 & 0.50 \\
\hline Revenue (\$) & 840 & 2520 & 3600 & 4080 & 3960 & 3240 & 1920 \\
\hline
\end{array}
</code></p><p>a) Create a scatter plot, and draw a quadratic curve of good fit.</p><p>b) Determine an equation in vertex form to model this relation.</p><p>c) Use your model to predict the revenue if the price of an ice cream cone is reduced to <code class='latex inline'>\displaystyle \$ 2.25 </code>.</p><p>d) To maximize revenue, what should an ice cream cone cost?</p><p>e) Check the accuracy of your model using quadratic regression.</p>

<p>A rectangle has a width of x centimetres, and its length is double its width.</p>
<ul>
<li>Create a table comparing the length and area of a rectangle for widths up to 8 cm.</li>
</ul>

<p>The scatter plot and curve of best fit show the relationship between the diameter of roan-collection barrels and the volume of water collected.
Is this relation linear or non-linear? Justify your answer. </p><img src="/qimages/750" />

<p>Determine, without graphing, which type of relationship (linear, quadratic, or neither) best models this table of values. Explain.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|r|r|r|r|r|r|}
\hline \boldsymbol{x} & -1 & 0 & 1 & 2 & 3 \\
\hline \boldsymbol{y} & 1 & 2 & -3 & -14 & -31 \\
\hline
\end{array}
</code></p>

<p>A hang-glider was launched form a platform on the top of a escarpment. The data describe the first 13s of the flight. The values for height are negative whenever the hang-glider was below the top of the escarpment.</p><p><strong>a)</strong> Determine the height of the platform.</p><p><strong>b)</strong> Determine an equation that models the height of the hang-glider over 13 s period.</p><p><strong>c)</strong> Determine the lowest height of the hang-glider and when it occurred.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline Time (s) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\
\hline Height (m) & 10.0 & -0.8 & -9.2 & -15.2 & -18.8 & -20.0 & -18.8 & -15.2 & -9.2 & -0.8 & 10.0 & 23.2 & 38.8 & 56.8 \\
\hline
\end{array}
</code></p>