9.3 intro to rational functions

9.3 Rational Functions and Their Graphs

A rational function is. . .

Examples of Rational Functions

Where does the denominator = 0?

Examples of Rational Functions

Where does the denominator = 0?

Examples of Rational Functions

Where does the denominator = 0?

A point of discontinuity occurs when. . .

It causes or

in the graph.

Example 1. Find the points of discontinuity.

a. 1x2x

1y

2 b.

1x

1xy

2

If a number is a solution to only the denominator of a rational function, then it creates an __________________ in the function.

ex.

)2x)(1x(

1xy

If a number is a solution to both the numerator and denominator of a rational function, then it creates a _____________ in the function.

ex.

)1x)(2x(

2xy

Example 2. Describe the vertical asymptotes and holes for the graph of each rational function.

a. b. )3x)(2x(

1xy

2x

)1x)(2x(y

c.

d.

)4x)(3x)(3x(

)4x)(3x(y

1x

1xy

2

Horizontal AsymptotesA horizontal asymptote depends on the

biggest power of the numerator and denominator.

1. If n > d,

2. If n < d,

3. If n = d,

Example 3. Find the horizontal asymptotes of each function.

a.

b.

c.

1x

6x2y

2x

1y

x5

x2xy

3

Example 4. Sketch the graph of

Points of discontinuity:

Holes:

Vertical Asymptotes:

Horizontal Asymptotes:

)4x)(3x(

2xy

Assignment: