Exam 3: Polynomial and Rational Functions

a. Identify the horizontal asymptote (if any). b. If the graph of the function has a horizontal asymptote, determine the point where the graph crosses the horizontal asymptote. - $f ( x ) = \frac { 7 } { x ^ { 2 } - 7 x + 9 }$

Determine if the graph can represent a polynomial function. If so, assume the end behavior and all turning points are represented on the graph. - a. Determine the minimum degree of the polynomial based on the number of turning points. b. Determine whether the leading coefficient is positive or negative based on the end behavior and Whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicity is odd or even.

Write the domain in interval notation and identify any vertical asymptotes. - $f ( x ) = \frac { x ^ { 2 } + x - 56 } { x - 7 }$

Identify the vertex and determine the minimum or maximum value of the function. - $f ( x ) = - 2 x ^ { 2 } + 8 x - 2$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. - $\text { The graph of } f ( x ) = \frac { 1 } { x + 6 } - 3 \text { is the graph of } y = \frac { 1 } { x } \text { shifted (left/right) } 6 \text { units and (up/down) } 3 \text { units. }$

Find the value of c that gives the function the given minimum value. - $f ( x ) = 5 x ^ { 2 } - 10 x + c ; \text { minimum value } 1$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -What is the leading term of $f ( x ) = - \frac { 1 } { 3 } ( x - 3 ) ^ { 4 } ( 3 x + 5 ) ^ { 2 }$ ?

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given the division algorithm identify the polynomials representing the dividend, divisor, quotient, and remainder. $f ( x ) = d ( x ) \cdot q ( x ) + r ( x )$

Find the zeros and their multiplicities. Consider using Descartes' rule of signs and the upper and lower bound theorem to limit your search for rational zeros. - $f ( x ) = 3 x ^ { 4 } - 32 x ^ { 3 } + 122 x ^ { 2 } - 188 x + 80$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Let $f ( x )$ be a rational expression. An inequality of the form $f ( x ) < 0 , f ( x ) > 0 , f ( x ) \geq 0$ , or $f ( x ) \leq 0$ is called aـــــــــــــــــ inequality.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If $C ( x )$ represents the cost to manufacture $x$ items, then $\bar { C } ( x ) =$ ـــــــــــــrepresents the average cost per item.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A rational function will have a slant asymptote if the degree of the numerator is exactly greater than the degree of the denominator.

Choose the one alternative that best completes the statement or answers the question. List the possible rational zeros. - $f ( x ) = 25 x ^ { 4 } - 7 x ^ { 3 } + 2 x + 10$

Solve the problem. -A rectangle is bounded by the x-axis and a parabola defined by $y = 4 - x ^ { 2 }$ . What are the dimensions of the rectangle if the area is $6 \mathrm {~cm} ^ { 2 }$ ? Assume that all units of length are in centimeters.

Solve the inequality. Write the solution set in interval notation. - $- 4 u ( u + 2 ) ^ { 2 } ( 5 - u ) > 0$

Solve the problem. -Determine if the upper bound theorem identifies 4 as an upper bound for the real zeros of f (x). $f ( x ) = 8 x ^ { 3 } - 8 x ^ { 2 } + 3 x - 9$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -a. Given y = 2x, evaluate y for the given values of x. $x = 1 , x = 2 , x = 3 , x = 4 , \text { and } x = 5 \text {. }$ b. How does y change when x is doubled? c. How does y change when x is tripled? d. Complete the statement. Given y = 2x, when x increases, y (increases/decreases) proportionally. e. Complete the statement. Given y = 2x, when x decreases, y (increases/decreases) proportionally.

$\text { The graph of } f ( x ) = a ( x - h ) ^ { 2 } + k , a \neq 0$ 0, is a parabola and the axis of symmetry is a line given by________

Choose the one alternative that best completes the statement or answers the question. Use long division to divide. - $\left( x ^ { 5 } + 2 x ^ { 4 } - 2 x ^ { 2 } - 5 x - 12 \right) \div \left( x ^ { 2 } - 3 \right)$

Identify the asymptotes. - $f ( x ) = \frac { 4 x ^ { 3 } + 3 x + 3 } { 2 x ^ { 2 } - 4 x + 3 }$