Exam 3: Polynomial and Rational Functions

Solve the problem. -The amount of water (in gallons) in a leaky bathtub is given in the table below. Using a graphing utility, fit the data to a third degree polynomial (or a cubic). Then approximate the time at which there is maximum amount Of water in the tub, and estimate the time when the water runs out of the tub. Express all your answers rounded To two decimal places. (in minutes) 0 1 2 3 4 5 6 7 ( in gallons) 20 26 45 63 86 94 90 67

Find the vertical asymptotes of the rational function. - $f ( x ) = \frac { x ( x - 1 ) } { x ^ { 3 } + 16 x }$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = 6 x ^ { 4 } + 4 x ^ { 3 } - 3 x ^ { 2 } + 2$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $x \text {-intercepts of } f ( x ) = \frac { x ^ { 3 } - 27 } { x ^ { 2 } - 25 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 5 ) ( x - 4 ) ( x + 4 )$

Solve the problem. -Decide which of the rational functions might have the given graph. A) $f ( x ) = \frac { 1 } { 2 x }$ B) $f ( x ) = x ^ { 2 }$ C) $f ( x ) = \frac { 1 } { x }$ D) $f ( x ) = \frac { 1 } { x ^ { 2 } }$

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis. - $f ( x ) = ( x + 15 ) ^ { 2 }$

Analyze the graph of the given function f as follows: (a) Determine the end behavior: find the power function that the graph of f resembles for large values of | $| x |$ (b) Find the x- and y-intercepts of the graph. (c) Determine whether the graph crosses or touches the x-axis at each x-intercept. (d) Graph f using a graphing utility. (e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places. (f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. - $f ( x ) = ( x - 3 ) ( x - 1 ) ( x + 2 )$

Find the domain of the rational function. - $g ( x ) = \frac { 5 x } { ( x + 2 ) ( x + 4 ) }$

Solve the problem. - $x ( x + 5 ) \geq - 6$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = - 2 x ^ { 3 } + 4 x ^ { 2 } - 3 x + 8$

Find all zeros of the function and write the polynomial as a product of linear factors. - $f ( x ) = x ^ { 3 } - x ^ { 2 } + 16 x - 16$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { 8 } { x ^ { 2 } - 3 x - 23 }$

Solve the problem. - $x ^ { 2 } - 8 x \geq 0$

Use the graph to find the horizontal asymptote, if any, of the function. -

Find the domain of the rational function. - $h ( x ) = \frac { x + 5 } { x ^ { 2 } - 36 }$

Find the vertical asymptotes of the rational function. - $h ( x ) = \frac { 6 x } { x + 2 }$

Solve the equation in the real number system. - $x ^ { 3 } + 3 x ^ { 2 } - 8 x + 10 = 0$

Solve the problem. -The concentration of a drug in the bloodstream, measured in milligrams per liter, can be modeled by the function, $C ( t ) = \frac { 12 t + 4 } { 3 t ^ { 2 } + 2 }$ where t is the number of minutes after injection of the drug. When will the drug be at its Highest concentration? Approximate your answer rounded to two decimal places.

Use the given zero to find the remaining zeros of the function. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39 \text {; zero: } - 3$