rational functions Asymptotes and domain and range

Ra#onal Func#ons In this lesson we will:

1. Review what a ra#onal func#on is

2. Determine the equa#ons for the asymptotes of a ra#onal func#on 3. Determine the domain and range of the func#on

1

Ra#onal Func#ons A ra#onal func#on is a frac#on with polynomials in the numerator and denominator

x−2 example : 2 x − 4x + 4 example :

3x − 4 2x + 4x − 5 2

2

Ra#onal Func#ons How do you find an asymptote?

Where does the graph not touch?

1. Graph the func#on.

2. Analyze the table or graph

example :

x−2 x 2 − 4x + 4

3

Ra#onal Func#ons

How do you find an asymptote?

Where does the graph not touch?

1. Graph the func#on.

graph

2. Analyze the table or

x−2 example : 2 x − 4x + 4

Asymptotes: the func#on is undefined At x=2, check the table the asymptote is x=2

Another asymptote is y=0, the graph never touches the x-­‐axis

3

Ra#onal Func#ons How do you find an asymptote? x=2, y=0

1. Graph the func#on.

2. Analyze the table or graph

x−2 example : 2 x − 4x + 4

the asymptotes are x=2, y=0

4

Ra#onal Func#ons How do you find the domain and range?

x−2 Asymptotes: x=2, y=0 example :

x 2 − 4x + 4

−∞

Domain

∞

The graph stretches from nega#ve infinity to infinity, except for x=2

D: All real numbers except 2

3

Ra#onal Func#ons How do you find the domain and range?

x−2 Asymptotes: x=2, y=0 example :

x 2 − 4x + 4

∞

Range

−∞ The graph stretches from nega#ve infinity to infinity, except for y=0

D: All real numbers except x=2 R: All real numbers except y=0

3

Kuta Software - Infinite Algebra 2

Name________________

Prac#ce Problems 1-­‐4. Determine the Asymptotes and Graphing Rational Functions Date_______ domain and range for each func#on

Identify the points of discontinuity, holes, vertical asymptotes, x-intercepts, and hori each.

1) f (x) =

3) f (x) =

x

2) f (x) =

x

x

x

x

x

x x

x x

4) f (x) =

x

x x

Identify the points of discontinuity, holes, vertical asymptotes, and horizontal asymp sketch the graph.

5) f (x) =

x

6) f (x) =

x

x x

y

y

x

7) f (x) =

x x

8) f (x) =

x x

x x

Prac#ce Problems holes, 5-­‐8. Dvertical etermine the Asymptotes and

Identify the points of discontinuity, asymptotes, and horizontal asymptote sketch the graph. domain and range for each func#on

5) f (x) =

x

6) f (x) =

x

x x

y

y

x

7) f (x) =

x

x

8) f (x) =

x y

x x

x x y

x

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x

-1-

9) f (x) =

Prac#ce Problems 9-­‐12. Determine the Asymptotes and domain and range for each func#on x x x x x

10) f (x) =

x

x

y

x y

x

11) f (x) =

x

x

x

12) f (x) =

x

x x

y

y

x

13) f (x) =

x x

x x y

x

14) f (x) =

x y

x

x

Socra#ve Problems 1-­‐4 Use the Func#ons below to answer the ques#ons on Socra#ve.

Determine the Asymptotes and domain and range for each func#on

x x

13) f (x) =

x

14) f (x) =

x y

x y

x

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x

-2-