Exam 3: Polynomial and Rational Functions

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }$

Graph the function. - $f(x)=x^{2}+\frac{1}{x}$

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

Use transformations of the graph o to graph the function. $y = x ^ { 4 } \text { or } y = x ^ { 5 }$ - $f(x)=(x+3)^{4}$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $f ( x ) = x ^ { 4 } + 11 x ^ { 3 } + 7 x ^ { 2 } + 71 x - 66 ; x - 11$

Use the x-intercepts to find the intervals on which the graph of f is above and below the x-axis. - $f ( x ) = ( x - 3 ) ^ { 3 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x - 4 ) ( x - 1 )$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 3 \right) ( x - 3 )$

Write the word or phrase that best completes each statement or answers the question. -a A rare species of insect was discovered in the rain forest of Costa Rica. Environmentalists transplant the insect in protected area. The population of the insect $t$ months after being transplanted is $\mathrm { P } ( \mathrm { t } ) = \frac { 45 ( 1 + 0.6 \mathrm { t } ) } { ( 3 + 0.02 \mathrm { t } ) }$ a) What was the population when $t = 0$ ? b) What will the population be after 10 years? c) What is the largest value the population could reach?

Give the equation of the horizontal asymptote, if any, of the function. - $R ( x ) = \frac { - 3 x ^ { 2 } } { x ^ { 2 } + 9 x - 22 }$

Find the domain of the rational function. - $f ( x ) = \frac { - 2 x ( x + 2 ) } { 5 x ^ { 2 } - 3 x - 8 }$

Write the word or phrase that best completes each statement or answers the question. 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. - $f ( x ) = - x ^ { 2 } ( x - 1 ) ( x + 3 )$

Find the x- and y-intercepts of f. - $f ( x ) = 2 x ^ { 4 } ( x - 1 ) ^ { 3 }$

Choose the one alternative that best completes the statement or answers the question. Find a bound on the real zeros of the polynomial function. - $f ( x ) = x ^ { 5 } + 3 x ^ { 4 } + 5 x ^ { 3 } - 2 x ^ { 2 } + 4 x - 1$

Use transformations of the graph o to graph the function. $y = x ^ { 4 } \text { or } y = x ^ { 5 }$ - $f(x)=(x+3)^{5}$

Write the word or phrase that best completes each statement or answers the question. -The profits (in millions) for a company for 8 years was as follows: Year, x Profits 1993,1 1.1 1994,2 1.7 1995,3 2.0 1996,4 1.4 1997,5 1.3 1998,6 1.5 1999,7 1.8 2000,8 2.1 Find the cubic function of best fit to the data.

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 4 ) \left( x ^ { 2 } + 1 \right)$

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 vertical asymptotes of the rational function. - $g ( x ) = \frac { x + 9 } { x ^ { 2 } + 4 }$

Use the Factor Theorem to determine whether x - c is a factor of f(x). - $9 x ^ { 4 } + 26 x ^ { 3 } - 3 x ^ { 2 } + x + 3 ; x + 3$