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Factor the polynomial completely: $x^3 - 4x^2 - 12x$ .

Select one option

A. $x(x-6)(x+2)$

B. $x(x+6)(x-2)$

C. $(x-6)(x+2)$

D. $x^2(x-4)-12x$

Describe the end behavior of the function $f(x) = -3x^4 + 2x^3 - 5x + 1$ .

Select one option

A. As $x \to \infty, f(x) \to \infty$ ; as $x \to -\infty, f(x) \to \infty$ .

B. As $x \to \infty, f(x) \to -\infty$ ; as $x \to -\infty, f(x) \to -\infty$ .

C. As $x \to \infty, f(x) \to \infty$ ; as $x \to -\infty, f(x) \to -\infty$ .

D. As $x \to \infty, f(x) \to -\infty$ ; as $x \to -\infty, f(x) \to \infty$ .

What are the real roots of the equation $x^3 - 9x = 0$ ?

Select one option

A. $0, 3$

B. $0, -3$

C. $0, 3, -3$

D. $3, -3$

Which of the following is equivalent to $2 \log_b x + \log_b y - \log_b z$?

Select one option

A. $\log_b \frac{x^2 y}{z}$

B. $\log_b \frac{2xy}{z}$

C. $\log_b (x^2 + y - z)$

D. $\log_b (x^2 y z)$

Evaluate $\log_4 64$ .

Select one option

A. $16$

B. $4$

C. $3$

D. $2$

What is the inverse function of $f(x) = 5^{x-2}$ ?

Select one option

A. $f^{-1}(x) = \log_5 x + 2$

B. $f^{-1}(x) = \log_5 (x+2)$

C. $f^{-1}(x) = \log_5 x - 2$

D. $f^{-1}(x) = 2 \log_5 x$

What is the domain of the function $f(x) = \frac{x+5}{x^2 - 4x - 12}$ ?

Select one option

A. All real numbers except $x=6$ and $x=-2$

B. All real numbers except $x=6$ and $x=2$

C. All real numbers except $x=-6$ and $x=2$

D. All real numbers except $x=4$ and $x=12$

Identify the vertical asymptotes of the function $g(x) = \frac{x-3}{x^2 - 9}$ .

Select one option

A. $x=3$ and $x=-3$

B. $x=-3$ only

C. $x=3$ only

D. No vertical asymptotes

Determine the horizontal asymptote of the function $h(x) = \frac{4x^2 - 1}{2x^2 + 5x + 3}$ .

Select one option

A. $y=0$

B. $y=2$

C. $y=\frac{1}{2}$

D. No horizontal asymptote

10.

Simplify the expression $\frac{x^2 - 16}{x^2 - 8x + 16}$ .

Select one option

A. $\frac{x+4}{x-4}$

B. $\frac{x-4}{x+4}$

C. $1$

D. $\frac{1}{x-4}$

11.

Which type of conic section is represented by the equation $4x^2 + y^2 - 8x + 4y - 8 = 0$?

Select one option

A. Parabola

B. Circle

C. Ellipse

D. Hyperbola

12.

What are the center and radius of the circle given by the equation $(x-3)^2 + (y+1)^2 = 25$ ?

Select one option

A. Center: $(3, -1)$ , Radius: $5$

B. Center: $(-3, 1)$ , Radius: $5$

C. Center: $(3, -1)$ , Radius: $25$

D. Center: $(-3, 1)$ , Radius: $25$

13.

Find the vertex of the parabola $y = 2x^2 - 12x + 15$ .

Select one option

A. $(3, -3)$

B. $(-3, 3)$

C. $(3, 15)$

D. $(-3, 15)$

14.

Which of the following is the equation of a hyperbola with vertices at $(0, \pm 4)$ and foci at $(0, \pm 5)$ ?

Select one option

A. $\frac{y^2}{16} - \frac{x^2}{9} = 1$

B. $\frac{x^2}{16} - \frac{y^2}{9} = 1$

C. $\frac{y^2}{25} - \frac{x^2}{16} = 1$

D. $\frac{x^2}{25} - \frac{y^2}{16} = 1$

15.

What is the exact value of $\sin\left(\frac{5\pi}{6}\right)$ ?

Select one option

A. $\frac{1}{2}$

B. $-\frac{1}{2}$

C. $\frac{\sqrt{3}}{2}$

D. $-\frac{\sqrt{3}}{2}$

16.

Simplify the expression $\frac{\tan x}{\sin x}$ .

Select one option

A. $\cos x$

B. $\sec x$

C. $\csc x$

D. $\cot x$

17.

Find all solutions for $2 \cos x - 1 = 0 $in the interval$ [0, 2\pi)$.

Select one option

A. $\frac{\pi}{3}, \frac{5\pi}{3}$

B. $\frac{\pi}{6}, \frac{11\pi}{6}$

C. $\frac{\pi}{3}, \frac{2\pi}{3}$

D. $\frac{2\pi}{3}, \frac{4\pi}{3}$

18.

The number of bacteria in a culture doubles every 3 hours. If there are 100 bacteria initially, which expression gives the number of bacteria after $t$ hours?

Select one option

A. $100(2)^{t/3}$

B. $100(2)^{3t}$

C. $100(3)^{t/2}$

D. $100(2t)^3$

19.

One root of the polynomial $P(x) = x^3 - 6x^2 + 3x + 10$ is $x=-1$ . What are the other real roots?

Select one option

A. $x=2, x=5$

B. $x=-2, x=5$

C. $x=2, x=-5$

D. $x=-2, x=-5$

20.

Which of the following rational functions has a slant asymptote?

Select one option

A. $f(x) = \frac{x^2 - 4}{x^3 + 1}$

B. $f(x) = \frac{x^2 - 4}{x - 1}$

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

D. $f(x) = \frac{x - 1}{x^2 - 4}$

21.

Consider the function $f(x) = \frac{x^2 - 4}{x^2 - x - 6}$ . a. Find the domain of the function. b. Identify any holes in the graph. c. Find the equations of all vertical asymptotes. d. Find the equation of the horizontal asymptote. e. Find the x-intercept(s) and y-intercept. f. Sketch the graph of the function, labeling all intercepts, asymptotes, and holes.

22.

Find all real and complex roots of the polynomial equation $x^4 - x^3 - 11x^2 + 9x + 18 = 0$ .

23.

A population of rabbits grows exponentially. There were 120 rabbits initially, and after 3 years, there are 500 rabbits. a. Write an exponential growth model for the rabbit population, $P(t)$ , where $t$ is in years. b. Estimate the number of rabbits after 5 years. c. How long will it take for the population to reach 2000 rabbits?

24.

A hyperbola has vertices at $(-3, 0)$ and $(3, 0)$ , and a focus at $(5, 0)$ . a. Find the equation of the hyperbola. b. Determine the equations of the asymptotes. c. Sketch the graph of the hyperbola, including its vertices, foci, and asymptotes.

25.

Prove the trigonometric identity: $\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x} = 2 \csc x$ .