Chapter 2

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=-x^{2}-4 x\)

The average annual price-earnings ratio for a corporation's stock is defined as the average price of the stock divided by the earnings per share. The average price of a corporation's stock is given as the function \(P\) and the earnings per share is given as the function \(E\). Find the price-earnings ratios, \(P / E\), for the years 2001 to 2005 . Jack in the Box $$\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 2001 & 2002 & 2003 & 2004 & 2005 \\\\\hline P & \$ 27.22 & \$ 28.19 & \$ 19.38 & \$ 25.20 & \$ 36.21 \\\\\hline E & \$ 2.11 & \$ 2.33 & \$ 2.04 & \$ 2.27 & \$ 2.48 \\\\\hline\end{array}$$

A company's weekly profit \(P\) (in hundreds of dollars) from a product is given by the model \(P(x)=80+20 x-0.5 x^{2}, \quad 0 \leq x \leq 20\) where \(x\) is the amount (in hundreds of dollars) spent on advertising. (a) Use a graphing utility to graph the profit function. (b) The company estimates that taxes and operating costs will increase by an average of $$\$ 2500$$ per week during the next year. Rewrite the profit equation to reflect this expected decrease in profits. Identify the type of transformation applied to the graph of the equation. (c) Rewrite the profit equation so that \(x\) measures advertising expenditures in dollars. [Find \(P(x / 100) .]\) Identify the type of transformation applied to the graph of the profit function.

Find the domain of the function. \(f(x)=\frac{x-4}{\sqrt{x}}\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{3}+2\)

Find the domains of \((f / g)(x)\) and \((g / f)(x)\) for the functions \(f(x)=\sqrt{x}\) and \(g(x)=\sqrt{9-x^{2}}\) Why do the two domains differ?

The number of horsepower \(H\) required to overcome wind drag on an automobile is approximated by \(H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100\) where \(x\) is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Rewrite the horsepower function so that \(x\) represents the speed in kilometers per hour. [Find \(H(x / 1.6) .]\) Identify the type of transformation applied to the graph of the horsepower function.

Find the domain of the function. \(f(x)=\frac{x-5}{\sqrt{x^{2}-9}}\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{3}-1\)

Use a graphing utility to graph the six functions below in the same viewing window. Describe any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

The change in volume \(V\) (in milliliters) of the lungs as they expand and contract during a breath can be approximated by the model $$V=\left(-6.549 s^{2}+26.20 s-3.8\right)^{2}, \quad 0 \leq s \leq 4$$ where \(s\) represents the number of seconds. Graph the volume function with a graphing utility and use the trace feature to estimate the number of seconds in which the volume is increasing and in which the volume is decreasing. Find the maximum change in volume between 0 and 4 seconds.

Consider \(f(x)=\sqrt{x-2}\) and \(g(x)=\sqrt[3]{x-2}\). Why are the domains of \(f\) and \(g\) different?

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=\frac{8}{x^{2}+4}\)

Determine whether the statement is true or false. Justify your answer. If you are given two functions \(f(x)\) and \(g(x)\), you can calculate \((f \circ g)(x)\) if and only if the range of \(g\) is a subset of the domain of \(f\).

Use the results of Exercise 67 to make a conjecture about the shapes of the graphs of \(y=x^{7}\) and \(y=x^{8} .\) Use a graphing utility to verify your conjecture.

For the years 1990 to 2005 , the book value \(B\) (in dollars) of a share of Wells Fargo stock can be approximated by the model \(B=0.0272 t^{2}+0.268 t+1.71, \quad 0 \leq t \leq 15\) where \(t\) represents the year, with \(t=0\) corresponding to 1990 (see figure). (Source: Wells Fargo) (a) Estimate the maximum book value per share from 1990 to 2005 . (b) Estimate the minimum book value per share from 1990 to 2005 . (c) Verify your estimates from parts (a) and (b) with a graphing utility.

A student says that the domain of \(f(x)=\frac{\sqrt{x+1}}{x-3}\) is all real numbers except \(x=3 .\) Is the student correct? Explain.

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=\frac{4}{x^{2}+1}\)

A unPower Corporation develops and manuAactures solar-electric power products. SunPower's new higher efficiency solar cells generate up to \(50 \%\) more power than other solar technologies. SunPower's technology was developed by Dr. Richard Swanson and his students while he was Professor of Engineering at Stanford University. SunPower's 2006 revenues are projected to increase \(300 \%\) from its 2005 revenues. Use your campus library, the Internet, or some other reference source to find information about an alternative energy business experiencing strong growth similar to the example above. Write a brief report about the company or small business.

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: y=3 x+4 ; L_{2}: y=x-\frac{1}{4}\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=\sqrt{x+1}\)

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: y=\frac{3}{4} x+1 ; L_{2}: y=-\frac{4}{3} x+3\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=\sqrt{1-x}\)

A company produces a product for which the variable cost is $$\$ 11.75$$ per unit and the fixed costs are $$\$ 112,000$$. The product sells for $$\$ 21.95$$ per unit. Let \(x\) be the number of units produced and sold. (a) Add the variable cost and the fixed costs to write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Use the formula \(P=R-C\) to write the profit \(P\) as a function of the number of units sold.

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: 2 x-y=1 ; L_{2}: x+2 y=-1\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=\sqrt[3]{x}\)

A company produces a product for which the variable cost is $$\$ 9.85$$ per unit and the fixed costs are $$\$ 85,000$$. The product sells for $$\$ 19.95$$ per unit. Let \(x\) be the number of units produced and sold. (a) Add the variable cost and the fixed costs to write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Use the formula \(P=R-C\) to write the profit \(P\) as a function of the number of units sold.

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: x-5 y=-2 ; L_{2}:-3 x+15 y=6\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. . \(y=\sqrt[3]{x+1}\)

The height \(y\) (in feet) of a baseball thrown by a child is given by $$y=-\frac{1}{10} x^{2}+3 x+6$$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: x-3 y=-3 ; L_{2}: 2 x-6 y=6\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=|x-4|\)

The cost of sending an overnight package from Los Angeles to Miami is $$\$ 10.75$$ for up to, but not including, the first pound and $$\$ 3.95$$ for each additional pound (or portion of a pound). A model for the total cost \(C\) of sending the package is \(C=10.75+3.95 \llbracket x \rrbracket, x>0\), where \(x\) is the weight of the package (in pounds). Sketch the graph of this function.

Part of the life cycle of a salmon is migration for reproduction. Salmon are anadromous fish. This means that they swim from the ocean to fresh water streams to lay their eggs. During migration, salmon must jump waterfalls to reach their destination. The path of a jumping salmon is given by \(h=-0.42 x^{2}+2.52 x\) where \(h\) is the height (in feet) and \(x\) is the horizontal distance (in feet) from where the salmon left the water. Will the salmon clear a waterfall that is 3 feet high if it leaves the water 4 feet from the waterfall?

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: 4 x-y=-2 ; L_{2}: 8 x-2 y=6\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=|x|-3\)

The total volume \(V\) (in millions of barrels) of the Strategic Oil Reserve \(R\) in the United States from 1995 to 2005 can be approximated by the model \(V=\left\\{\begin{array}{ll}-2.722 t^{3}+61.18 t^{2}-451.5 t+1660, & 5 \leq t \leq 10 \\ 34.7 t+179, & 11 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. Sketch the graph of this function. (Source: U.S. Energy Information Administration)

The national defense budget expenses for veterans \(V\) (in billions of dollars) in the United States from 1990 to 2005 can be approximated by the model \(V=\left\\{\begin{array}{ll}-0.326 t^{2}+3.40 t+28.7, & 0 \leq t \leq 6 \\\ 0.441 t^{2}-6.23 t+62.6, & 7 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=0\) corresponding to 1990\. Use the model to find total veteran expenses in 1995 and 2005. (Source: U.S. Office of Management and Budget)

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: 2 x-3 y-15=0 ; L_{2}: 3 x+2 y+8=0\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x=y^{2}-1\)

The 2007 salary \(S\) (in dollars) for federal employees at the Step 1 level can be approximated by the model \(S=\left\\{\begin{array}{ll}2904.3 x+12,155, & x=1,2, \ldots, 10 \\ 11,499.2 x-81,008, & x=11, \ldots .15\end{array}\right.\) where \(x\) represents the "GS" grade. Sketch a bar graph that represents this function. (Source: U.S. Office of Personnel Management)

The number \(N\) (in thousands) of mobile homes manufactured for residential use in the United States from 1991 to 2005 can be approximated by the model \(N=\left\\{\begin{array}{ll}29.08 t+157.0, & 1 \leq t \leq 7 \\ 4.902 t^{2}-151.70 t+1289.2, & 8 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=1\) corresponding to 1991\. Use the model to find the total number of mobile homes manufactured between 1991 and \(2005 .\) Source: U.S. Census Bureau)

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: x-4 y-12=0 ; L_{2}: 3 x-4 y-8=0\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x=y^{2}-4\)

The total numbers (in thousands) of U.S. airline delays, cancellations, and diversions for the years 1995 to 2005 are given by the following ordered pairs. (Source: U.S. Bureau of Transportation Statistics) \((1995,5327.4)(1996,5352.0)(1997,5411.8)\) \((1998,5384.7)(1999,5527.9)(2000,5683.0)\) \((2001,5967.8)(2002,5271.4)(2003,6488.5)\) \((2004,7129.3)(2005,7140.6)\) (a) Use the regression feature of a graphing utility to find a quadratic model for the data from 1995 to 2001 . Let \(t\) represent the year, with \(t=5\) corresponding to \(1995 .\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data from 2002 to 2005 . Let \(t\) represent the year, with \(t=12\) corresponding to 2002 . (c) Use your results from parts (a) and (b) to construct a piecewise model for all of the data.

The total sales \(S\) (in millions of dollars) for the Cheesecake Factory for the years 1999 to 2005 are shown in the table. (Source: Cheesecake Factory) $$\begin{array}{|l|l|l|l|l|}\hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\\\\hline \text { Sales, } S & 347.5 & 438.3 & 539.1 & 652.0 \\\\\hline\end{array}$$ $$\begin{array}{|l|c|c|c|}\hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Sales, } S & 773.8 & 969.2 & 1177.6 \\\\\hline\end{array}$$ (a) Use a graphing utility to create a scatter plot of the data. Let \(\mathrm{t}\) represent the year, with \(t=9\) corresponding to \(1999 .\) (b) Use the regression feature of a graphing utility to find a linear model and a quadratic model for the data. (c) Use each model to approximate the total sales for each year from 1999 to 2005 . Compare the values generated by each model with the actual values shown in the table. Which model is a better fit? Justify your answer.

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-5,0),(-2,1) ; L_{2}:(0,1),(3,2)\)

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x^{2}+y^{2}=4\)

The revenues of Symantec Corporation (in millions of dollars) from 1996 to 2005 are given by the following ordered pairs. (Source: Symantec Corporation) \((1996,472.2)(1997,578.4)(1998,633.8)\) \((1999,745.7)(2000,853.6)(2001,1071.4)\) \((2002,1406.9)(2003,1870.1)(2004,2582.8)\) \((2005,4143.4)\) (a) Use the regression feature of a graphing utility to find a linear model for the data from 1996 to 2000 . Let \(t\) represent the year, with \(t=6\) corresponding to 1996 . (b) Use the regression feature of a graphing utility to find a quadratic model for the data from 2001 to \(2005 .\) Let \(t\) represent the year, with \(t=11\) corresponding to 2001 . (c) Use your results from parts (a) and (b) to construct a piecewise model for all of the data.

The book values per share \(B\) (in dollars) for Analog Devices for the years 1996 to 2005 are shown in the table. (Source: Analog Devices) $$\begin{array}{|c|c|}\hline \text { Year } & \text { BV/share, } B \\ \hline 1996 & 2.72 \\\\\hline 1997 & 3.36 \\\\\hline 1998 & 3.52 \\ \hline 1999 & 4.62 \\\\\hline 2000 & 6.44 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|} \hline \text { Year } & \text { BV/share, } B \\\\\hline 2001 & 7.83 \\\\\hline 2002 & 7.99 \\\\\hline 2003 & 8.88 \\ \hline 2004 & 10.11 \\\\\hline 2005 & 10.06 \\\\\hline\end{array}$$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to \(1996 .\) (b) Use the regression feature of a graphing utility to find a linear model and a quadratic model for the data. (c) Use each model to approximate the book value per share for each year from 1996 to \(2005 .\) Compare the values generated by each model with the actual values shown in the table. Which model is a better fit? Justify your answer.