Chapter 1

Solve each problem. Women against the Men The men's Olympic pole vaulting winning heights in meters during year \(x\) can be approximated by \(H(x)=\frac{1}{48} x-35.83,\) where \(1896 \leq x \leq 2008\) (Assume that \(x\) is a multiple of 4 because the Olympics occur every 4 years.) (a) Evaluate \(H(1920)\) and interpret the result. (b) In 2008 the women's Olympic winning height in the pole vault was about 5 meters. Determine the years when this height would have beaten or tied the men's winning heights.

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{7}{3}(2 x-1)=\frac{1}{5} x+\frac{2}{5}(4-3 x)$$

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-2,-4)$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{c|c|c|c|c} x & 11 & 12 & 13 & 14 \\ \hline y & -6 & -6 & -7 & -6 \end{array}$$

Graph each line by hand. Give the \(x\)- and y-intercepts. \(2 x+5 y=10\)

Solve each problem. Online Holiday Shopping In 2011 , online holiday sales were \(\$ 192\) billion, and in 2014 , they were \(\$ 249\) billion. (Source: Digital Lifestyles.) (a) Find a linear function \(S\) that models these data, where \(x\) is the year. (b) Interpret the slope of the graph of \(S\). (c) Predict when online holiday sales might reach \(\$ 325\) billion.

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$0.1 x-0.05=-0.07 x$$

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-2,4)$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{c|c|c|c|c} \boldsymbol{x} & 1 & 1 & 1 & 1 \\ \hline \boldsymbol{y} & 12 & 13 & 14 & 15 \end{array}$$

Graph each line by hand. Give the \(x\)- and y-intercepts. \(4 x-3 y=9\)

Solve each problem. Bicycle Safety \(\quad\) A survey found that \(76 \%\) of bicycle riders do not wear helmets. (Source: Opinion Research Corporation for Glaxo Wellcome, Inc.) (a) Write a linear function \(f\) that computes the number of people who do not wear helmets among \(x\) bicycle riders. (b) There are approximately 38.7 million riders of all ages who do not wear helmets. Write a linear equation whose solution gives the total number of bicycle riders. Find this number of riders.

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$1.1 x-2.5=0.3(x-2)$$

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(3,0)$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & \sqrt{2} & \sqrt{3} & \sqrt{5} & \sqrt{6} & \sqrt{7} \end{array}$$

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=3 x$$

Modeling I Exercises \(33-36\) (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(R\) as a function of \(x .\) (c) Determine analytically the value of \(x\) for which revenue equals cost. (d) Graph \(y_{1}=C(x)\) and \(y_{2}=R(x)\) on the same \(x y\) -axes and interpret the graphs. Stuffing Envelopes\(\quad\) A student stuffs envelopes for extra income during her spare time. Her initial cost to obtain the necessary information for the job was \(\$ 200.00 .\) Each envelope costs \(\$ 0.02,\) and she gets paid \(\$ 0.04\) per envelope stuffed. Let \(x\) represent the number of envelopes stuffed.

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$0.40 x+0.60(100-x)=0.45(100)$$

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-2,0)$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\begin{array}{c|c|c|c|c|c} \boldsymbol{x} & 1 & \frac{1}{2} & \frac{1}{4} & \frac{1}{8} & \frac{1}{16} \\\ \hline \boldsymbol{y} & 0 & -1 & -2 & -3 & -4 \end{array}$$

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=-2 x$$

Exercises \(33-36\) (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(R\) as a function of \(x .\) (c) Determine analytically the value of \(x\) for which revenue equals cost. (d) Graph \(y_{1}=C(x)\) and \(y_{2}=R(x)\) on the same \(x y\) -axes and interpret the graphs. Copier Service \(\quad\) A technician runs a copying service in his home. He paid \(\$ 3500\) for the copier and a lifetime service contract. Each sheet of paper costs \(\$ 0.01,\) and he gets paid \(\$ 0.05\) per copy. Let \(x\) be the number of copies he makes.

Answer each question. (a) What is the equation of the \(x\) -axis? (b) What is the equation of the \(y\) -axis?

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$1.30 x+0.90(0.50-x)=1.00(50)$$

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(3,-3)$$

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=-0.75 x$$

Exercises \(33-36\) (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(R\) as a function of \(x .\) (c) Determine analytically the value of \(x\) for which revenue equals cost. (d) Graph \(y_{1}=C(x)\) and \(y_{2}=R(x)\) on the same \(x y\) -axes and interpret the graphs. Delivery Service \(A\) truck driver operates a delivery Service. His start-up costs amounted to \(\$ 2300 .\) He estimates that it costs him (in terms of gasoline, wear and tear eon his truck, etc.) \(\$ 3.00\) per delivery. He charges \(\$ 5.50\) per delivery. Let \(x\) represent the number of deliveries he makes.

Graph each linear function on a graphing calculator, using the two different windows given. State which window gives a comprehensive graph. $$f(x)=4 x+20$$ Window A: \([-10,10]\) by \([-10,10]\) Window B: \([-10,10]\) by \([-5,25]\)

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$2[x-(4+2 x)+3]=2 x+2$$

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$x y>0$$

A line having an equation of the form \(y=k x,\) where \(k\) is a real number, \(k \neq 0,\) will always pass through the origin. To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=1.5 x$$

\(f(x)=-5 x+30\) Window A: \([-10,10]\) by \([-10,40]\) Window B: \([-5,5]\) by \([-5,40]\)

Exercises \(33-36\) (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(R\) as a function of \(x .\) (c) Determine analytically the value of \(x\) for which revenue equals cost. (d) Graph \(y_{1}=C(x)\) and \(y_{2}=R(x)\) on the same \(x y\) -axes and interpret the graphs. Baking and Selling Cakes\(\quad\) A baker makes cakes and sells them at county fairs. Her initial cost for the Pointe Coupee parish fair was \(\$ 40.00 .\) She figures that each cake costs \(\$ 2.50\) to make, and she charges \(\$ 6.50\) per cake. Let \(x\) represent the number of cakes sold. (Assume that there were no cakes left over.)

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$6[x-(2-3 x)+1]=4 x-6$$

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$x y<0$$

Write equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. \(5 x+3 y=15\) \([-10,10]\) by \([-10,10]\)

Graph each linear function on a graphing calculator, using the two different windows given. State which window gives a comprehensive graph. $$f(x)=3 x+10$$ Window \(A:[-3,3]\) by \([-5,5]\) Window B: \([-5,5]\) by \([-10,14]\)

In Exercises \(37-40,\) find the constant of variation \(k\) and the undetermined value in the table if \(y\) is directly proportional to \(x\). $$\begin{array}{c|c|c|c|c} x & 3 & 5 & 6 & 8 \\ \hline y & 7.5 & 12.5 & 15 & ? \end{array}$$

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{5}{6} x-2 x+\frac{1}{3}=\frac{1}{3}$$

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$\frac{x}{y}<0$$

Write equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. \(6 x+5 y=9\) \([-10,10]\) by \([-10,10]\)

Graph each linear function on a graphing calculator, using the two different windows given. State which window gives a comprehensive graph. $$f(x)=3 x+10$$ Window \(A:[-3,3]\) by \([-5,5]\) Window B: \([-5,5]\) by \([-10,14]\)

In Exercises \(37-40,\) find the constant of variation \(k\) and the undetermined value in the table if \(y\) is directly proportional to \(x\). $$\begin{array}{c|c|c|c|c} x & 1.2 & 4.3 & 5.7 & ? \\ \hline y & 3.96 & 14.19 & 18.81 & 23.43 \end{array}$$

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{3}{4}+\frac{1}{5} x-\frac{1}{2}=\frac{4}{5} x$$

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$\frac{x}{y}>0$$

Find the slope (if defined) of the line that passes through the given points. \((\)-2,1)\( and \)(3,6)$$

Write equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. \(-2 x+7 y=4\) \([-5,5]\) by \([-5,5]\)

In Exercises \(37-40,\) find the constant of variation \(k\) and the undetermined value in the table if \(y\) is directly proportional to \(x\). Sales tax \(y\) on a purchase of \(x\) dollars \begin{tabular}{c|c|c|c} \(\boldsymbol{x}\) & \(\$ 25\) & \(\$ 55\) & \(?\) \\ \hline \(\boldsymbol{y}\) & \(\$ 1.50\) & \(\mathfrak{S} 3.30\) & \(\mathfrak{S} 5.10\) \end{tabular}

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$5 x-(8-x)=2[-4-(3+5 x-13)]$$

If the \(x\) -coordinate of a point is \(0,\) the point must lie on which axis?

Find the slope (if defined) of the line that passes through the given points. $$(-2,3)\( and \)(-1,2) \quad$$