Question 1

Given the rational function $$R(x) = \frac{x-1}{x^2$-9}$$, which of the following statements about its asymptotes is correct?

Accepted Answer

There are vertical asymptotes at $$x = -3$$ and $$x = 3$$, and a horizontal asymptote at $$y = 0.$$

Question 2

For the rational function $$R(x) = \frac{x^2$-4}{x}$$, what is the equation of the oblique asymptote?

Accepted Answer

$$y = x$$

Question 3

A rational function has a numerator of degree 2 and a denominator of degree 2. What type of asymptote will the graph have?

Accepted Answer

A horizontal asymptote at $$y = \frac{a}{b}$$, where $$a$$ and $$b$$ are the leading coefficients of the numerator and denominator, respectively.

Question 4

Which of the following best describes the behavior of the graph of $$R(x) = \frac{x^4$+1}{x^2$} as$$ $$x 	o \infty$$?

Accepted Answer

The graph approaches $$y = x^2$.

Question 5

Suppose $$R(x) = \frac{2x^2$ - 2x}{x^2$ + x - 6}. $$What are the $$x-$$intercepts of the graph of $$R$$?

Accepted Answer

$$x = 0$$ and $$x = 1$$

Question 6

A rational function $$R(x)$$ has a hole at $$x = 3$$ and a vertical asymptote at $$x = -3. $$Which of the following could be the denominator of $$R(x) in $$lowest terms?

Accepted Answer

$$(x+3)$$

Question 7

Given the cost function for manufacturing a cylindrical can with fixed volume $$V = 650 cm$^3$$, $$C(r) = 2\pi r^2$ + \frac{1300}{r}$$, what is the real-world meaning of the minimum point on the graph of $$C(r)$$?

Accepted Answer

It is the radius that minimizes the cost of materials for the can.

Question 8

If a rational function $$R(x)$$ has a vertical asymptote at $$x = 2 of $$even multiplicity, what does this tell you about the behavior of the graph near $$x = 2$$?

Accepted Answer

The graph approaches the same infinity on both sides of $$x = 2.$$

Question 9

Which of the following is a necessary step when graphing a rational function $$R(x)$$?

Accepted Answer

Find the zeros of the numerator and denominator.

Question 10

A rational function $$R(x)$$ has a horizontal asymptote at $$y = 2. $$What does this tell you about the degrees and leading coefficients of the numerator and denominator?

Accepted Answer

The degrees are equal, and the ratio of leading coefficients is 2.

Question 11

Which of the following best describes the x-intercepts of a rational function $$R(x) = \frac{(x+1)^2$(x-3)}{(x+4)(x-2)^2$}$$?

Accepted Answer

$$x = -1 ($$multiplicity 2) and $$x = 3 ($$multiplicity 1)

Question 12

If the graph of a rational function $$R(x)$$ touches the x-axis at $$x = -1$$ but does not cross it, what can you conclude about the factor $$(x+1) in $$the numerator?

Accepted Answer

It has even multiplicity.

Question 13

A rational function $$R(x)$$ has a hole at $$x = 3. $$How do you find the y-coordinate of the hole?

Accepted Answer

Evaluate the simplified function at $$x = 3$$ after canceling the common factor.

Question 14

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

Accepted Answer

$$R(x) = \frac{x^2$+1}{x^3$-2}$$

Question 15

Suppose you are designing a can with fixed volume $$V$$ and want to minimize the cost of materials. If the cost function is $$C(r) = 2\pi r^2$ + \frac{1300}{r}$$, what mathematical technique would you use to find the optimal radius?

Accepted Answer

Find the minimum of $$C(r) by $$taking the derivative and setting it to zero.

Question 16

If $$R(x) = \frac{(x-1)(x+2)}{(x-3)^2$}$$, what is the behavior of the graph near $$x = 3$$?

Accepted Answer

The graph approaches $$+\infty on $$both sides of $$x = 3.$$

Question 17

Which of the following is true about the intersection of a rational function's graph with its horizontal asymptote?

Accepted Answer

The graph may intersect the horizontal asymptote at one or more points.

Graphing and Analyzing Rational Functions

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