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

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

Q: Given $f(x) = \dfrac{1}{x-1}$, which of the following statements is true?

The function has a vertical asymptote at $x = 1$

Q: A parabola has its vertex at $(2, -3)$ and opens upward. Which of the following could be its equation?

$y = a(x - 2)^2 - 3$, where $a > 0$

Question 1

A company models its profit $P$ (in thousands of dollars) as a function of time $t$ (in years) by the equation $P(t) = 5t^2 - 20t + 30$. What is the minimum profit, and after how many years does it occur?

Accepted Answer

The minimum profit is $10$ thousand dollars, occurring at $t = 2$ years.

Question 2

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

Accepted Answer

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

Question 3

A biologist models the population of a bacteria culture by $P(t) = 200e^{0.3t}$, where $t$ is in hours. How long will it take for the population to triple?

Accepted Answer

$t = \dfrac{\ln 3}{0.3}$ hours

Question 4

Which of the following is the correct inverse of the function $f(x) = 2x - 5$?

Accepted Answer

$f^{-1}(x) = \dfrac{x + 5}{2}$

Question 5

A rational function is given by $f(x) = \dfrac{3x^2 - 2x + 1}{x^2 - 4}$. Which of the following describes its vertical asymptotes?

Accepted Answer

Vertical asymptotes at $x = 2$ and $x = -2$

Question 6

A ball is thrown upward from the top of a 50-meter building with an initial velocity of 20 m/s. Its height $h$ (in meters) after $t$ seconds is given by $h(t) = -4.9t^2 + 20t + 50$. After how many seconds does the ball reach its maximum height?

Accepted Answer

$t = 2.04$ seconds

Question 7

Which of the following equations represents a circle with center at $(3, -2)$ and radius $5$?

Accepted Answer

$(x - 3)^2 + (y + 2)^2 = 25$

Question 8

Given the function $f(x) = \sqrt{x - 4}$, what is the domain of $f$?

Accepted Answer

$x \geq 4$

Question 9

A radioactive substance decays according to $A(t) = 100e^{-0.05t}$, where $A$ is the amount remaining after $t$ years. What is the half-life of the substance?

Accepted Answer

$t = \dfrac{\ln 2}{0.05}$ years

Question 10

Which of the following best describes the graph of $y = |x - 3| + 2$?

Accepted Answer

A V-shaped graph with vertex at $(3, 2)$

Question 11

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

Accepted Answer

$2x^2 + 1$

Question 12

A car's value depreciates exponentially according to $V(t) = 25000(0.85)^t$, where $t$ is the number of years after purchase. What is the value of the car after 3 years?

Accepted Answer

$V(3) = 25000 	imes (0.85)^3$

Question 13

Which of the following is the solution set for the inequality $2x - 5 < 7$?

Accepted Answer

$x < 6$

Question 14

Given $f(x) = \dfrac{1}{x-1}$, which of the following statements is true?

Accepted Answer

The function has a vertical asymptote at $x = 1$

Question 15

A quadratic equation $ax^2 + bx + c = 0$ has real and equal roots if:

Accepted Answer

$b^2 - 4ac = 0$

Question 16

Which of the following is the solution to the equation $\log_2(x) = 5$?

Accepted Answer

$x = 32$

Question 17

A parabola has its vertex at $(2, -3)$ and opens upward. Which of the following could be its equation?

Accepted Answer

$y = a(x - 2)^2 - 3$, where $a > 0$

Question 18

Which of the following is the correct equation for the line passing through the points $(1, 2)$ and $(3, 6)$?

Accepted Answer

$y = 2x$

Question 19

If $f(x) = x^2 - 4x + 7$, what is the minimum value of $f(x)$?

Accepted Answer

$3$

Question 20

Which of the following is the solution to the system of equations: $\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$?

Accepted Answer

$x = 2, y = 5$

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