Exam 3: Polynomial and Rational Functions

Solve the problem. -Determine which rational function $R ( x )$ has a graph that crosses the $x$ -axis at $- 1$ , touches the $x$ -axis at $- 4$ , has vertical asymptotes at $x = - 2$ and $x = 3$ , and has one horizontal asymptote at $y = - 2$ .

Graph the function. - $f ( x ) = \frac { x ^ { 2 } - 5 x + 4 } { ( x - 5 ) ^ { 2 } }$ A) B) C) D)

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

Use the given zero to find the remaining zeros of the function. - $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 42 x ^ { 3 } - 124 x ^ { 2 } + 297 x - 306$ ; zero: $3 i$

Use the graph to find the vertical asymptotes, if any, of the function. -

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = x ^ { 3 / 2 } - x ^ { 6 } + 7$

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. - $f ( x ) = ( x + 3 ) ( x - 2 ) ^ { 2 }$

Solve the problem. - $( x - 1 ) \left( x ^ { 2 } + x + 1 \right) >$

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - $f ( x ) = x ^ { 6 } - 12 x ^ { 4 } - 69 x ^ { 2 } + 80 ; c = 4$

Solve the equation in the real number system. - $2 x ^ { 3 } - x ^ { 2 } - 12 x + 6 = 0$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $7 x ^ { 3 } + 17 x ^ { 2 } - 11 x + 3 ; x + 3$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = 7 x ^ { 3 } - x ^ { 2 } + 2$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { x - 5 } { 3 x - 3 }$

Use transformations of the graph o $y = x ^ { 4 } \text { or } y = x ^ { 5 }$ to graph the function. - $f ( x ) = ( x - 3 ) ^ { 5 } + 2$ A) B) C) D)

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. - $x ^ { 4 } - x ^ { 3 } - 7 x ^ { 2 } + 5 x + 10 = 0 ; - 3 \leq r \leq - 2$

Graph the function. - $f ( x ) = \frac { x ^ { 4 } - 1 } { x ^ { 2 } - 16 }$ A) B) C) D)

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = x ^ { 5 } - 3 x ^ { 2 } + 3 x + 35$

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. -Degree 3; zeros: $4,1 - \mathrm { i }$

Solve the problem. -The acceleration due to gravity g (in meters per second per second) at a height h meters above sea level is given by $g ( h ) = \frac { 3.99 \times 10 ^ { 14 } } { \left( 6.374 \times 10 ^ { 6 } + h \right) ^ { 2 } }$ where $6.374 \times 10 ^ { 6 }$ is the radius of Earth in meters. Death Valley in California is 86 m below sea level. a) Find the value of g(h) at Death Valley to four decimal places. b) Compare the value in (a) to the value of g(h) at sea level.

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { x ^ { 3 } + 15 } { x ^ { 2 } - 5 }$