Chapter 3

The average number of daily phone calls, \(C\), between two cities varies jointly as the product of their populations, \(P_{1}\) and \(P_{2}\) and inversely as the square of the distance, \(d\), between them. a. Write an equation that expresses this relationship. b. The distance between San Francisco (population: \(777,000\) ) and Los Angeles (population: \(3,695,000\) ) is 420 miles. If the average number of daily phone calls between the cities is \(326,000,\) find the value of \(k\) to two decimal places and write the equation of variation. c. Memphis (population: \(650,000\) ) is 400 miles from New Orleans (population: \(490,000\) ). Find the average number of daily phone calls, to the nearest whole number, between these cities.

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{12 x}{3 x^{2}+1}$$

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)-2 x^{4}-5 x^{3}-x^{2}-6 x+4 $$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=x^{3}+x^{2}-2 x+1 ; \text { between }-3 \text { and }-2 $$

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=x^{4}+5 x^{3}+5 x^{2}-5 x-6 ; f(3) $$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}+7 x^{2}-x-7<0 $$

The force of wind blowing on a window positioned at a right angle to the direction of the wind varies jointly as the area of the window and the square of the wind's speed. It is known that a wind of 30 miles per hour blowing on a window measuring 4 feet by 5 feet exerts a force of 150 pounds. During a storm with winds of 60 miles per hour, should hurricane shutters be placed on a window that measures 3 feet by 4 feet and is capable of withstanding 300 pounds of force?

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{15 x}{3 x^{2}+1}$$

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)-4 x^{4}-x^{3}+5 x^{2}-2 x-6 $$

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=x^{4}-5 x^{3}+5 x^{2}+5 x-6 ; f(2) $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-6-4 x+x^{2}$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}+x^{2}+4 x+4>0 $$

The table shows the values for the current, \(I,\) in an electric circuit and the resistance, \(R,\) of the circuit. $$\begin{array}{|l|l|l|l|l|l|l|l|l|}\hline I \text { (amperes) } & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 4.0 & 5.0 \\ \hline R \text { (ohms) } & 12.0 & 6.0 & 4.0 & 3.0 & 2.4 & 2.0 & 1.5 & 1.2 \\\\\hline\end{array}$$ a. Graph the ordered pairs in the table of values, with values of \(I\) along the \(x\) -axis and values of \(R\) along the \(y\) -axis. Connect the eight points with a smooth curve. b. Does current vary directly or inversely as resistance? Use your graph and explain how you arrived at your answer. c. Write an equation of variation for \(I\) and \(R\), using one of the ordered pairs in the table to find the constant of variation. Then use your variation equation to verify the other seven ordered pairs in the table.

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$g(x)=\frac{12 x^{2}}{3 x^{2}+1}$$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=3 x^{3}-10 x+9 ; \text { between }-3 \text { and }-2 $$

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=2 x^{4}-5 x^{3}-x^{2}+3 x+2 ; \quad f\left(-\frac{1}{2}\right) $$

In Exercises \(39-44,\) an equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)-3 x^{2}-12 x-1$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}-x^{2}+9 x-9>0 $$

What does it mean if two quantities vary directly?

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$g(x)=\frac{15 x^{2}}{3 x^{2}+1}$$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ f(x)-x^{3}-4 x^{2}-7 x+10 $$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=3 x^{3}-8 x^{2}+x+2 ; \text { between } 2 \text { and } 3 $$

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=6 x^{4}+10 x^{3}+5 x^{2}+x+1 ; \quad f\left(-\frac{2}{3}\right) $$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3} \geq 9 x^{2} $$

In your own words, explain how to solve a variation problem.

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$h(x)=\frac{12 x^{3}}{3 x^{2}+1}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{3}+2 x^{2}-x-2 $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 2 x^{3}-x^{2}-9 x-4-0 $$

Use synthetic division to divide $$ f(x)=x^{3}-4 x^{2}+x+6 \text { by } x+1 $$ Use the result to find all zeros of \(f\)

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)--4 x^{2}+8 x-3$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3} \leq 4 x^{2} $$

What does it mean if two quantities vary inversely?

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$h(x)=\frac{15 x^{3}}{3 x^{2}+1}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{3}+x^{2}-4 x-4 $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ 3 x^{3}-8 x^{2}-8 x+8-0 $$

Use synthetic division to divide $$ f(x)=x^{3}-4 x^{2}+x+6 \text { by } x+1 $$ Use the result to find all zeros of \(f\)

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)--2 x^{2}-12 x+3$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x-4}{x+3}>0 $$

Explain what is meant by combined variation. Give an example with your explanation.

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{-2 x+1}{3 x+5}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{4}-9 x^{2} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ f(x)-x^{4}-2 x^{3}+x^{2}+12 x+8 $$

Solve the equation \(2 x^{3}-5 x^{2}+x+2=0\) given that 2 is a zero of \(f(x)=2 x^{3}-5 x^{2}+x+2\)

An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)-5 x^{2}-5 x$$

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+5}{x-2}>0 $$

Explain what is meant by joint variation. Give an example with your explanation.

Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{-3 x+7}{5 x-2}$$

A. Use the Leading Coefficient Test to determine the graph's end behavior. B. Find the \(x\) -intercepts. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each intercept. C. Find the \(y\) -intercept. D. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. E. If necessary, find a few additional points and graph the function. Use the maximum number of uning points to check whether it is drawn correctly. $$ f(x)=x^{4}-x^{2} $$

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ f(x)-x^{4}-4 x^{3}-x^{2}+14 x+10 $$

Solve the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) given that \(-2\) is a zero of \(f(x)=2 x^{3}-3 x^{2}-11 x+6\)