Chapter 5

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{2 x^{3}+25}{x+3} $$

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. \(\begin{array}{|c|c|c|c|c|c|}\hline x & {-2} & {-1} & {0} & {1} & {2} \\\ \hline y & {8} & {2} & {0} & {2} & {8} \\ \hline\end{array}\)

Make a table to confirm the end behavior of the function. $$f(x)=\frac{x^{5}}{10}-x^{4}$$

For the following exercises, use the given information to answer the questions. The distance \(s\) that an object falls varies directly with the square of the time, \(t,\) of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

For the following exercises, find the inverse of the functions with \(a, b, c\) positive real numbers. $$f(x)=a x^{3}+b$$

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=2 x^{3}+37 x^{2}+200 x+300\)

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=5\) and \(x=-5, x\) -intercepts at \((2,0)\) and \((-1,0), y\) -intercept at \((0,4)\)

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x^{3}(x-2) $$

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{3 x^{3}+2 x-5}{x-1} $$

For the following exercises, use a calculator to find the answer. Graph on the same set of axes the functions \(f(x)=x^{2}, f(x)=2 x^{2},\) and \(f(x)=\frac{1}{3} x^{2} .\) What appears to be the effect of changing the coefficient?

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}(x-2)$$

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=x^{3}-2 x^{2}-16 x+32\)

For the following exercises, use the given information to answer the questions. The velocity \(v\) of a falling object varies directly to the time, \(t,\) of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=-4\) and \(x=-1, x\) -intercepts at \((1,0)\) and \((5,0), y\) -intercept at \((0,7)\)

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x(x-3)(x+3) $$

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{-4 x^{3}-x^{2}-12}{x+4} $$

For the following exercises, use a calculator to find the answer. Graph on the same set of axes \(f(x)=x^{2}, f(x)=x^{2}+2\) and \(f(x)=x^{2}, f(x)=x^{2}+5\) and \(f(x)=x^{2}-3 .\) What appears to be the effect of adding a constant?

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(x-3)(x+3)$$

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=2 x^{4}-5 x^{3}-5 x^{2}+5 x+3\)

For the following exercises, use the given information to answer the questions. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=-4\) and \(x=-5, x\) -intercepts at \((4,0)\) and \((-6,0),\) horizontal asymptote at \(y=7\)

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x(14-2 x)(10-2 x) $$

For the following exercises, find the inverse of the functions with \(a, b, c\) positive real numbers. \(f(x)=\sqrt{a x^{2}+b}\)

For the following exercises, use synthetic division to find the quotient and remainder. $$ \frac{x^{4}-22}{x+2} $$

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(14-2 x)(10-2 x)$$

For the following exercises, find the inverse of the functions with \(a,\) c positive real numbers. $$f(x)=\sqrt[3]{a x+b}$$

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=2 x^{4}-5 x^{3}-14 x^{2}+20 x+8\)

For the following exercises, use the given information to answer the questions. The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=-3\) and \(x=6, x\) -intercepts at \((-2,0)\) and \((1,0),\) horizontal asymptote at \(y=-2\)

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x(14-2 x)(10-2 x) $$

For the following exercises, use a calculator with CAS to answer the questions. Consider \(\frac{x^{k}-1}{x-1}\) with \(k=1,2,3 .\) What do you expect the result to be if \(k=4 ?\)

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(14-2 x)(10-2 x)^{2}$$

For the following exercises, find the inverse of the functions with \(a, b, c\) positive real numbers. $$f(x)=\frac{a x+b}{x+c}$$

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=10 x^{4}-21 x^{2}+11\)

For the following exercises, use the given information to answer the questions. The weight of an object above the surface of the Earth varies inversely with the square of the distance from the center of the Earth. If a body weighs 50 pounds when it is 3960 miles from Earth's center, what would it weigh it were 3970 miles from Earth's center?

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptote at \(x=-1\) double zero at \(x=2, y\) -intercept at \((0,2)\)

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x^{3}-16 x $$

For the following exercises, use a calculator with CAS to answer the questions. Consider \(\frac{x^{k}+1}{x+1}\) for \(k=1,3,5 .\) What do you expect the result to be if \(k=7 ?\)

For the following exercises, use a calculator to find the answer. A suspension bridge can be modeled by the quadratic function \(h(x)=0.0001 x^{2}\) with \(-2000 \leq x \leq 2000\) where \(|x|\) is the number of feet from the center and \(h(x)\) is height in feet. Use the [TRACE] feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-16 x$$

For the following exercises, determine the function described and then use it to answer the question. An object dropped from a height of 200 meters has a height, \(h(t)\) , in meters after \(t\) seconds have lapsed, such that \(h(t)=200-4.9 t^{2}\) . Express tas a function of height, \(h\) , and find the time to reach a height of 50 meters.

For the following exercises, list all possible rational zeros for the functions. \(f(x)=x^{4}+3 x^{3}-4 x+4\)

For the following exercises, use the given information to answer the questions. The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 footcandles at a distance of 3 meters. Find the intensity level at 8 meters.

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptote at \(x=3\) double zero at \(x=1, y\) -intercept at \((0,4)\)

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x^{3}-27 $$

For the following exercises, use a calculator with CAS to answer the questions. Consider \(\frac{x^{4}-k^{4}}{x-k}\) for \(k=1,2,3 .\) What do you expect the result to be if \(k=4 ?\)

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. Vertex \((1,-2),\) opens up.

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-27$$

For the following exercises, list all possible rational zeros for the functions. \(f(x)=2 x^{3}+3 x^{2}-8 x+5\)

For the following exercises, determine the function described and then use it to answer the question. An object dropped from a height of 600 feet has a height, \(h(t),\) in feet after \(t\) seconds have elapsed, such that \(h(t)=600-16 t^{2}\) . Express \(t\) as a function of height \(h\) , and find the time to reach a height of 400 feet.