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

$$f(0) = -1  0$$

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

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