Q: A biologist is studying the population of bacteria in a petri dish, which doubles every 3 hours. If the initial population is 200, which function best models the population $$P(t)$$ after $$t$$ hours?

$$P(t) = 200 \times 2$$^{t/3}$$

Q: Which of the following is the end behavior of the polynomial $$f(x) = -3x^5$ + 2x^3$ - x + 7$$?

$$f(x) \to -\infty as$$ $$x \to \infty$$; $$f(x) \to \infty as$$ $$x \to -\infty$$

Q: A quadratic function models the height $$h (in $$meters) of a ball thrown upward from a bridge as $$h(t) = -4.9t^2$ + 14t + 30. $$After how many seconds does the ball reach its maximum height?

$$t = \frac{-b}{2a} = \frac{-14}{2 \times -4.9} \approx 1.43$$ seconds

Q: Which of the following best describes the end behavior of the rational function $$f(x) = \frac{2x^3$ + 5}{x^3$ - 4}$$?

$$f(x) \to 2 as$$ $$x \to \infty$$ and $$x \to -\infty$$

Question 1

A biologist is studying the population of bacteria in a petri dish, which doubles every 3 hours. If the initial population is 200, which function best models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 	imes 2$$^{t/3}$$

Question 2

Given the quadratic function $$f(x) = 2x^2$ - 8x + 6$$, what is the vertex of the parabola?

Accepted Answer

$$(2, -2)$$

Question 3

A car's value $$V (in $$thousands of dollars) depreciates according to the function $$V(t) = 25(0.85)^t$$, where $$t is $$the number of years since purchase. What is the average rate of change in value from year 2 to year 5?

Accepted Answer

$$V(5) - V(2) \over 5-2 = {25(0.85)^5$ - 25(0.85)^2$ \over 3} \approx -2.44$$

Question 4

Which of the following is the end behavior of the polynomial $$f(x) = -3x^5$ + 2x^3$ - x + 7$$?

Accepted Answer

$$f(x) 	o -\infty as$$ $$x 	o \infty$$; $$f(x) 	o \infty as$$ $$x 	o -\infty$$

Question 5

Which of the following rational functions has a vertical asymptote at $$x = 2$$ and a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$f(x) = \frac{3}{x-2}$$

Question 6

A company models its profit $$P(x) in $$thousands of dollars as $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold (in hundreds). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$P(3) = 2$$ thousand dollars at $$x = 3 (i.e$$., 300 units)

Question 7

Which of the following is equivalent to $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$ for $$x 
eq 3$$?

Accepted Answer

$$\frac{x+3}{x-3}$$

Question 8

If $$f(x) = 2x^2$ - 5x + 3$$ and $$g(x) = x - 1$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$2(x-1)^2 - 5(x-1) + 3$

Question 9

A rational function $$f(x)$$ has a hole at $$x = 4$$ and a vertical asymptote at $$x = -2. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = \frac{(x-4)(x+1)}{(x-4)(x+2)}$$

Question 10

Which of the following statements about the function $$f(x) = \frac{2x^2$ + 3x - 2}{x^2$ - 1} is $$true?

Accepted Answer

It has vertical asymptotes at $$x = 1$$ and $$x = -1.$$

Question 11

Given $$f(x) = x^3$ - 4x^2$ + 5x - 2$$, which of the following is a complex zero of $$f(x)$$?

Accepted Answer

$$1 + i$$

Question 12

A quadratic function models the height $$h (in $$meters) of a ball thrown upward from a bridge as $$h(t) = -4.9t^2$ + 14t + 30. $$After how many seconds does the ball reach its maximum height?

Accepted Answer

$$t = \frac{-b}{2a} = \frac{-14}{2 	imes -4.9} \approx 1.43$$ seconds

Question 13

Which of the following best describes the end behavior of the rational function $$f(x) = \frac{2x^3$ + 5}{x^3$ - 4}$$?

Accepted Answer

$$f(x) 	o 2 as$$ $$x 	o \infty$$ and $$x 	o -\infty$$

Question 14

Which transformation maps the graph of $$f(x) = x^2$ to g(x$$) = $$-2(x-3)^2 + 5$?

Accepted Answer

Shift right 3 units, vertically stretch by 2, reflect over $$x-$$axis, shift up 5 units

Question 15

A function $$f(x) is $$defined as $$f(x) = \frac{3x^2$ - 12}{x^2$ - 4}. $$For which value(s) of $$x$$ does $$f(x)$$ have a hole?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 16

Which of the following is the correct partial fraction decomposition of $$\frac{5x+3}{(x+1)(x-2)}$$?

Accepted Answer

$$\frac{2}{x+1} + \frac{3}{x-2}$$

Question 17

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 1}{x - 1}. $$What is the simplified form of $$f(x)$$ for $$x 
eq 1$$?

Accepted Answer

$$x + 1$$

Question 18

If $$f(x) = 3x^2$ - 2x + 1$$ and $$g(x) = x - 4$$, what is $$f(g(x))$$?

Accepted Answer

$$3(x-4)^2 - 2(x-4) + 1$

Question 19

Which of the following is a valid assumption when modeling the height of a projectile with a quadratic function?

Accepted Answer

Air resistance is negligible and gravity is constant.

Question 20

A chemical reaction's rate is modeled by $$R(t) = 5e$$^{-0.2t}$$, where t$ is time in minutes. What is the rate after 10 minutes?

Accepted Answer

$$R(10) = 5e$$^{-2}$$ \approx 0.68$$

Polynomial and Rational Functions: Unit Overview and Key Skills

Study Guide - Practice Questions

Study Guide - Flashcards