Exam 4: Polynomial and Rational Functions

Solve the problem -Your company uses the quadratic model $y = - 11 x ^ { 2 } + 350 x$ to represent how many units (y) of a new product will be sold ( $\mathrm { x } )$ weeks after its release. How many units can you expect to sell in week $20 ?$

Find the equation that the given graph represents. - $f ( x ) = - \frac { 1 } { 3 } x ^ { 3 }$

Solve the problem -The number of mosquitoes $\mathrm { M } ( \mathrm { x } )$ , in millions, in a certain area depends on the June rainfall $\mathrm { x }$ , in inches, according to the equation $\mathrm { M } ( \mathrm { x } ) = 9 \mathrm { x } - \mathrm { x } ^ { 2 }$ . What rainfall produces the maximum number of mosquitoes?

Identify the vertex of the parabola. - $f ( x ) = - 5 ( x - 4 ) ^ { 2 } + 2$

Find an equation for the rational function graph. -As one approaches a certain sound source, the sound intensity in arbitrary units increases according to the following function of time: $F ( x ) = - \frac { 2 ( x - 3 ) ^ { 2 } + 3 ( x - 3 ) - 5 } { 2 ( x - 3 ) + 30 }$ . Starting at $x = 0$ , wher the source of sound is first sensed, how much time elapses until maximum sound intensity is felt? Round the result to the nearest hundredth.

Provide an appropriate response. -Find a quadratic function f having x-intercepts -9 and -2 and y-intercept 6.

Sketch the graph of the rational function. - $f ( x ) = \frac { 2 x } { x ^ { 2 } + 6 x + 5 }$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial $f ( x ) = x ^ { 4 } - x ^ { 3 } + 2 x ^ { 2 } - 4 x - 10$ has no real zero less than $- 1$ .

Provide an appropriate response. -Why is it not possible for a function defined by a polynomial of degree 4 with real coefficients to have zeros of 2, 3, 5, and 1 + i?

Identify any vertical, horizontal, or oblique asymptotes in the graph of y = f(x). State the domain of f. -

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros of $- 3 , - 1,4$ and $\mathrm { P } ( 2 ) = 5$

Determine which of the rational functions given below has the following feature(s). -There is a "hole" in its graph at x = 3

Provide an appropriate response. -Find a quadratic function f having x-intercepts 3 and -4 and y-intercept -24.

Find all rational zeros and factor f(x). - $f ( x ) = x ^ { 3 } + 8 x ^ { 2 } - 16 x - 128$

Find the equation that the given graph represents. - $f ( x ) = ( x + 3 ) ^ { 3 } - 4$

Give the domain and range for the rational function. Use interval notation. - $f ( x ) = \frac { 1 } { ( x + 7 ) ^ { 2 } }$

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zeros of $- 5 , i , - i$ and $P ( - 3 ) = 60$

Give the equation of any oblique asymptotes. - $h ( x ) = \frac { 5 } { x - 7 }$

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 7 x - 5$

Answer the question - $\text { How can the graph of } f ( x ) = - \frac { 4 } { x } \text { be obtained from the graph of } y = \frac { 1 } { x } ?$