Chapter 3

In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ where \(\frac{p}{q}\) is a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as $$ \begin{array}{c}a_{n} p^{n}+a_{n-1} p^{n-1} q \\\\+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}\end{array} $$ b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{p}{q}\) is reduced to lowest terms, \(p\) cannot divide \(q .\) Thus, \(p\) and \(q\) have no common factors other than \(-1\) and \(1 .\) Because \(p\) does divide the right side and it is not a factor of \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n^{*}}\)

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}-1}{x^{2}-9}$$

Explain why a polynomial function of degree 20 cannot cross the \(x\)-axis exactly once.

A company is planning to manufacture mountain bikes. Fixed monthly cost will be \(\$ 100,000\) and it will cost \(\$ 100\) to produce each bicycle. a. Write the cost function, \(C,\) of producing \(x\) mountain bikes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) mountain bikes. c. Find and interpret \(\bar{C}(500), \bar{C}(1000), \bar{C}(2000),\) and \(\bar{C}(4000)\) d. What is the horizontal asymptote for the function, \(\bar{C} ?\) Describe what this means in practical terms.

Give an example of a function that is not subject to the Intermediate Value Theorem.

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000 .\) It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C,\) of producing \(x\) pairs of shoes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), \quad \bar{C}(10,000), \quad\) and \(\bar{C}(100,000)\) d. What is the horizontal asymptote for the average cost function, \(\bar{C} ?\) Describe what this represents for the company.

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is the same as the given point. \(f(x)=3(x+2)^{2}-5 ; \quad(-1,-2)\)

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is the same as the given point. \(f(x)=(x-3)^{2}+2 ; \quad(6,11)\)

The rational function $$C(x)=\frac{130 x}{100-x}, 0 \leq x<100$$ describes the cost, \(C(x),\) in millions of dollars, to inoculate \(x \%\) of the population against a particular strain of flu. a. Find and interpret \(C(20), C(40), C(60), C(80),\) and \(C(90)\) b. What is the equation of the vertical asymptote? What does this mean in terms of the variables in the function? c. Graph the function.

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by a quadratic function. Group members should select the two sets of data that are most interesting and relevant. For each data set selected: a. Use the quadratic regression feature of a graphing utility to find the quadratic function that best fits the data. b. Use the equation of the quadratic function to make a prediction from the data. What circumstances might affect the accuracy of your prediction? c. Use the equation of the quadratic function to write and solve a problem involving maximizing or minimizing the function.

Among all deaths from a particular disease, the percentage that are smoking related ( \(21-39\) cigarettes per day) is a function of the disease's incidence ratio. The incidence ratio describes the number of times more likely smokers are than nonsmokers to die from the disease. The following table shows the incidence ratios for heart disease and lung cancer for two age groups. Incidence Ratios $$\begin{array}{|l|cc|} \hline & \text { Heart Disease } & \text { Lung Cancer } \\ \hline \text { Ages } 55-64 & 1.9 & 10 \\ \text { Ages } 65-74 & 1.7 & 9 \\ \hline \end{array}$$ For example, the incidence ratio of 9 in the table means that smokers between the ages of 65 and 74 are 9 times more likely than nonsmokers in the same group to die from lung cancer. The rational function $$P(x)=\frac{100(x-1)}{x}$$ models the percentage of smoking-related deaths among all deaths from a disease, \(P(x),\) in terms of the disease's incidence ratio, \(x\). The graph of the rational function is shown. Use this function to solve Exercises . (graph can't copy) Find \(P(9) .\) Round to the nearest percent. Describe what this means in terms of the incidence ratio, 9 given in the table. Identify your solution as a point on the graph.

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=x^{3}+13 x^{2}+10 x-4$$

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{4}+8 x^{3}+4 x^{2}+2$$

What is a rational function?

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-x^{5}+5 x^{4}-6 x^{3}+2 x+20$$

In Exercises \(78-79,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \(200 \mathrm{M} \text { OUT }]\) feature to show that \(f\) and g have identical end behavior. $$f(x)=x^{3}-6 x+1, \quad g(x)=x^{3}$$

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

If you are given the equation of a rational function, explain how to find the vertical asymptotes, if any, of the functions graph.

Which one of the following is true? a. If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and to the right. b. A mathematical model that is a polynomial of degree \(n\) whose leading term is \(a_{n} x^{n}, n\) odd and \(a_{n}<0,\) is ideally suited to describe nonnegative phenomena over unlimited periods of time. c. There is more than one third-degree polynomial function with the same three \(x\) -intercepts. d. The graph of a function with origin symmetry can rise to the left and to the right.

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the functions graph.

Describe how to graph a rational function.

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does how do you find its equation?

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Use a graphing utility to graph \(y=\frac{1}{x}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Use a graphing utility to graph $$f(x)=\frac{x^{2}-4 x+3}{x-2} \quad \text { and} \quad g(x)=\frac{x^{2}-5 x+6}{x-2}$$ What differences do you observe between the graph of \(f\) and \(g ?\) How do you account for these differences?

The rational function $$f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\) a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

Which one of the following is true? a. The graph of a rational function cannot have both a vertical and a horizontal asymptote. b. It is not possible to have a rational function whose graph has no \(y\) -intercept. c. The graph of a rational function can have three horizontal asymptotes. d. The graph of a rational function can never cross a vertical asymptote.

Which one of the following is true? a. The function \(f(x)=\frac{1}{\sqrt{x-3}}\) is a rational function. b. The \(x\) -axis is a horizontal asymptote for the graph of $$f(x)=\frac{4 x-1}{x+3}.$$ c. The number of televisions that a company can produce per week after \(t\) weeks of production is given by $$N(t)=\frac{3000 t^{2}+30,000 t}{t^{2}+10 t+25}.$$ Using this model, the company will eventually be able to produce \(30,000\) televisions in a single week. d. None of the given statements is true.

In Exercises \(93-96\), write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=3,\) a horizontal asymptote \(y=0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2, a\) horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=1,\) a slant Esymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\) -intercepts at \(-1\) and 2.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has no vertical. horizontal, or slant asymptotes, and no \(x\) -intercepts.