Section 2.6 Rational FunctionsPart 1

What you should learn

• How to find the domains of rational functions• How to find the horizontal and vertical

asymptotes of graphs of rational functions• How to analyze and sketch graphs of rationale

functions• How to sketch graphs of rational functions that

have slant asymptotes• How to use rational functions to model real-

life problems

Rational Function

A rational function can be written in the form

Where N(x) and D(x) are polynomials.

D(x) is not the zero polynomial.

)(

)()(

xD

xNxf

Find the domain of f(x)

158

1)(

2

xxxf

1580 2 xx)5)(3(0 xx

}5,3{ x

}5,3,|{ xxx

Find the domain of g(x)

2

1)(

xxg

20 x}2{x

}2,|{ xxx

2

-2

5

Horizontal Asymptote at y = 0 when degree of numerator less than degree of the denominator.

Vertical Asymptote at the zero of the denominator.

Asymptotes

• Vertical at zeros of the denominator

• HorizontalN < D asymptote at y = 0N = D asymptote at y = aN / aD

N > D no Horizontal asymptote

)(

)(

rdenominato degree

numerator degree

xD

xN

N = D asymptote at y = aN / aD

6

4

2

-5 5

y=f(x)

2

2 53)(

x

xxf

1

3y0x

Guidelines for Analyzing the Graphs of Rational Functions

1. Find and plot the y-intercept by evaluating f(0).

2. Find the zeros of the numerator and plot the points.

3. Find the zeros of the denominator and sketch the vertical asymptotes.

4. Find and sketch the horizontal asymptote if it exists.

Guidelines for Analyzing the Graphs of Rational Functions

5. Test for symmetry

6. Plot at least one point between and one point beyond each x intercept and vertical asymptote.

7. Use smooth curves to complete the graph between and beyond the vertical asymptotes

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

1. y-intercept when x = 0

f(0) = undefined

x = 0 is a vertical asymptote

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

2. Find the zeros of the numerator

0 = x2 – 4

x = {-2, 2}

= (x + 2)(x – 2)

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

3. Find the zeros of the Denominator

0 = x2 x = {0}

We already found this in step #1

x = 0 is a vertical asymptote

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

4. Find and sketch the horizontal asymptote

Since the degree of the numerator and the denominator are the same the horizontal asymptote will be y = 1/1=1

y = 1

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

5. Test for Symmetry

Since f(x)= f(-x) we know this drawing will be symmetrical about the y-axis

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

6. Plot some points

f(1)= f(-1) = -3

f(3)= f(-3) = 5/9

Graph

2

-2

-4

-5 5

2

2 4)(

x

xxf

7. Use smooth curves to complete the graph.

2

-2

-4

-5 5

y=f(x)

2.6 Homework Part 1

• Rational Functions

• page 174

• 5 - 39 odd,