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3.5: Rational Functions

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1. Rational Functions (Definition)

is a quotient of polynomials \dfrac{P(x)}{Q(x)}.

are all Rational Functions

\{x | x \text{ is not }1 \text{ or } -1\}.

2. Vertical Asymptotes

of a rational function occurs where the denominator is 0.

f(x) \rightarrow\infty.

Four Types of Vertical Asymptotes

Below are the four types of vertical asymptotes:

3. Horizontal Asymptotes

and we have the the left and right behavior of the graph is a horizontal line y = 3.

4. Oblique Asymptotes

Consider the function

f(x) = \dfrac{(x^2 - 3x - 4)}{(x + 3)}

alt

f(x) does not have a horizontal asymptote, since

\dfrac{x^2}{x}= x

is not a constant, but we see (on the calculator) that the left and right behavior of the curve is like a line. Our goal is to find the equation of this line.

We use synthetic division to see that

\dfrac{(x^2 - 3x - 4)}{(x + 3)} = x - 6 + \dfrac{14}{(x+3)}.

For very large x,

\dfrac{14}{x} + 3

is very small, hence f(x) is approximately equal to

x - 6

on the far left and far right of the graph. We call this line an Oblique Asymptote.

To graph, we see that there is a vertical asymptote at

x = -3

with behavior:

left down and right up

The graph has x-intercepts at 4 and -1, and a y intercept at -\frac{4}{3}.

\dfrac{(x^3 + 8)}{(x^2 - 3x - 4)}

5. Rational Functions With Common Factors

Consider the graph of

y = \dfrac{x-1}{x-1}

alt

What is wrong with the picture? When

f(x) = \dfrac{g(x)(x - r)}{h(x)(x - r)}

with neither g(r) nor h(r) zero, the graph will have a hole at x = r. We call this hole a removable discontinuity.

at (2,2).

We end our discussion with a list of steps for graphing rational functions.

Steps in graphing rational functions:

  • Step 1 Plug in x = 0 to find the y-intercept
  • Step 2 Factor the numerator and denominator. Cancel any common factors remember to put in the appropriate holes if necessary.
  • Step 3 Set the numerator = 0 to find the x-intercepts
  • Step 4 Set the denominator = 0 to find the vertical asymptotes. Then plug in nearby values to fine the left and right behavior of the vertical asymptotes.
  • Step 5 If the degree of the numerator = degree of the denominator then the graph has a horizontal asymptote. To determine the value of the horizontal asymptote, divide the term highest power of the numerator by the term of highest power of the denominator.
  • If the degree of the numerator = degree of the denominator + 1, then use polynomial or synthetic division to determine the equation of the oblique asymptote.
  • Step 6 Graph it!

Larry Green (Lake Tahoe Community College)

  • Integrated by Justin Marshall.


This page titled 3.5: Rational Functions is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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