Question 1

For the rational function $f(x) = \frac{x^2 + 16}{x^2 - 25}$, which of the following correctly lists the equations for the vertical and horizontal asymptotes?

Accepted Answer

Vertical asymptotes: $x = 5$, $x = -5$; Horizontal asymptote: $y = 1$

Question 2

Given the graph of a rational function with a vertical asymptote at $x = -4$ and a horizontal asymptote at $y = 4$, what is the domain of the function?

Accepted Answer

$(-\infty, -4) \cup (-4, \infty)$

Question 3

Use long division to divide the polynomials: $(5x^4 - 13x^3 - 13x^2 + 22x - 2) \div (x - 3)$. What is the quotient?

Accepted Answer

$5x^3 + 2x^2 - 7x + 1 + \frac{1}{x-3}$

Question 4

Determine the end behavior of the polynomial $f(x) = 19x^3 - 9x^2 + 2x - 8$ using the Leading Coefficient Test.

Accepted Answer

Graph falls to the left and rises to the right

Question 5

A quadratic function has a vertex at $(8, 4)$ and one zero at $x = 4$. What is the other zero?

Accepted Answer

$x = 12$

Question 6

Using Descartes' Rule of Signs, how many possible positive and negative real zeros does $f(x) = 6x^4 - x^3 - 4x^2 - 2x + 9$ have?

Accepted Answer

2 or 0 positive real zeros, 2 or 0 negative real zeros

Question 7

Express that the function $f(x) = 4x^3 + 9x^2 + 2x - 1$ has a real zero between $x = 0$ and $x = 2$ using the Intermediate Value Theorem.

Accepted Answer

$f(0) = -1 < 0; f(2) = 71 > 0$

Question 8

Given the graph of a rational function with a vertical asymptote at $x = 5$ and a zero at $x = 4$, which of the following is the correct formula for $f(x)$?

Accepted Answer

$f(x) = \frac{5(x-4)}{x-5}$

Question 9

Express $\log_{7} 343 = 6$ in its equivalent exponential form.

Accepted Answer

$7^6 = 343$

Question 10

Solve for $x$ in the equation $x^2 = 4913$.

Accepted Answer

$x = 17$

Question 11

Given $g(x) = e^{5x}$, what is the domain, range, and horizontal asymptote of the function?

Accepted Answer

Domain: $(-\infty, \infty)$; Range: $(0, \infty)$; Horizontal asymptote: $y = 0$

Question 12

Solve the logarithmic equation for $x$: $\log_4 (x^2 - 225) - \log_4 (x + 15) = 1$.

Accepted Answer

$x = 19$

Question 13

Expand the logarithmic expression $\log \left( \frac{100 \sqrt{2-y}}{3(y+8)^3} \right)$ using properties of logarithms.

Accepted Answer

$2 + \frac{1}{2} \log(2-y) - \log 3 - 3\log(y+8)$

Question 14

A chemical reaction rate is modeled by $R(t) = \frac{100}{t^2 + 4}$, where $t$ is time in seconds. What is the horizontal asymptote of $R(t)$ and what does it represent physically?

Accepted Answer

Horizontal asymptote: $y = 0$; as time increases, the reaction rate approaches zero.

Question 15

A population of bacteria doubles every hour and is modeled by $P(t) = 500 \cdot 2^t$. What is the population after 5 hours?

Accepted Answer

$P(5) = 16,000$

Question 16

A quadratic function models the height $h$ (in meters) of a ball thrown upward: $h(t) = -4.9t^2 + 20t + 2$. At what time $t$ does the ball reach its maximum height?

Accepted Answer

$t = 2.04$ seconds

Question 17

A medical dosage is modeled by $D(x) = \frac{50x}{x^2 + 25}$, where $x$ is the number of hours after administration. What is the domain of $D(x)$?

Accepted Answer

$(-\infty, \infty)$

Question 18

A company’s profit is modeled by $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of units sold. What is the maximum profit and at how many units sold does it occur?

Accepted Answer

Maximum profit: $P(10) = 100$ at $x = 10$ units

College Algebra Study Notes: Rational Functions, Polynomials, Zeros, and Logarithms

Study Guide - Practice Questions

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