Chapter 1

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Use point plotting to graph \(f(x)=x+2\) if \(x \leq 1\)

What must be done to a function's equation so that its graph is stretched vertically?

Simplify: \(2(x+h)^{2}+3(x+h)+5-\left(2 x^{2}+3 x+5\right)\)

What must be done to a function's equation so that its graph is shrunk horizontally?

I graphed $$f(x)=\left\\{\begin{array}{lll} 2 & \text { if } & x \neq 4 \\ 3 & \text { if } & x=4 \end{array}\right.$$ and one piece of my graph is a single point.

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f\left(\frac{1}{2} x\right),\) and \(h(x)=f\left(\frac{1}{4} x\right)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g,\) and \(h,\) with emphasis on different values of \(x\) for points on all three graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(0

Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)

During the winter, you program your home thermostat so that at midnight, the temperature is \(55^{\circ} .\) This temperature is maintained until 6 a. \(M\). Then the house begins to warm up so that by 9 A.M. the temperature is \(65^{\circ} .\) At 6 P.M. the house begins to cool. By 9 P.M., the temperature is again \(55^{\circ}\). The graph illustrates home temperature, \(f(t),\) as a function of hours after midnight, \(t\). (Graph can't copy) Using the graph at the bottom of the previous column, determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain \([0,24] .\) If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain [0,24] I decided to keep the house \(5^{\circ}\) warmer than before, so I reprogrammed the thermostat to \(y=f(t)+5\)

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.

During the winter, you program your home thermostat so that at midnight, the temperature is \(55^{\circ} .\) This temperature is maintained until 6 a. \(M\). Then the house begins to warm up so that by 9 A.M. the temperature is \(65^{\circ} .\) At 6 P.M. the house begins to cool. By 9 P.M., the temperature is again \(55^{\circ}\). The graph illustrates home temperature, \(f(t),\) as a function of hours after midnight, \(t\). (Graph can't copy) Using the graph at the bottom of the previous column, determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain \([0,24] .\) If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain [0,24] I decided to keep the house \(5^{\circ}\) cooler than before, so I reprogrammed the thermostat to \(y=f(t)-5\)

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even,odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h \quad\) definitely an odd function?

You invested \(\$ 80,000\) in two accounts paying \(5 \%\) and \(7 \%\) annual interest. If the total interest earned for the year was \(\$ 5200,\) how much was invested at each rate? (Section \(\mathrm{P.8}\) Example 5 )

Solve for \(A: C=A+A r\) (Section P.7, Example 5 )

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of the graph of \(f\) three units to the right and three units upward.

Solve by the quadratic formula: \(5 x^{2}-6 x-8=0\) (Section P.7, Example 10)

Will help you prepare for the material covered in the next section. $$\text { If }\left(x_{1}, y_{1}\right)=(-3,1) \text { and }\left(x_{2}, y_{2}\right)=(-2,4), \text { find } \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$

Will help you prepare for the material covered in the next section. Find the ordered pairs \((\quad, 0)\) and \((0, \quad)\) satisfying \(4 x-3 y-6=0\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting about the \(x\) -axis, and then moving the resulting graph down four units.

Will help you prepare for the material covered in the next section. Solve for \(y: 3 x+2 y-4=0\)

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$y=2 f(x)$$

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$y=f(x)-3$$

The length of a rectangle exceeds the width by 13 yards. If the perimeter of the rectangle is 82 yards, what are its dimensions?

Solve: \(\sqrt{x+10}-4=x\)

If \(f(x)=x^{2}+3 x+2,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) and simplify.

Exercises \(156-158\) will help you prepare for the material covered in the next section. Perform the indicated operation or operations. $$(2 x-1)\left(x^{2}+x-2\right)$$

Exercises \(156-158\) will help you prepare for the material covered in the next section. Perform the indicated operation or operations. $$(f(x))^{2}-2 f(x)+6, \text { where } f(x)=3 x-4$$

Exercises \(156-158\) will help you prepare for the material covered in the next section. Simplify: \(\frac{2}{\frac{3}{x}-1}\)