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-