Definition of a Rational Function

N(x)f(x)

D(x)

Any function of the form

Where N(x) and D(x) are polynomials andD(x) is not the zero polynomial

Examples.3x 5

f(x)x 4

24x 3x 5f(x)

5x 1

3x 5f(x) not a rational function

7

1 ,

1 ,

as x y

and

as x y

x=1 is a vertical asymptote because _____

2 1( )

1

xf x

x

Vertical asymptote

at x = 1

1 ,

1 ,

as x y

and

as x y

x=1 is a vertical asymptote because_____

2

1( )

1f x

x

Vertical asymptote

at x = 1

( )( )

( )N x

f xD x

vertical asymptote ( set denominator = 0 of reduced fraction) x + 5 = 0 x = -5

2

5Given: ( )

25

xf x

x

There is a vertical asymptote at x = -5

5

5 5

x

x x

1

5x

There is a hole at x = 5 ( the zeros of common factors)

2

5( )

25

xf x

x

Graph of Graph of

Find all vertical asymptotes and holes of 2 1

( )2

xf x

x

x = 2 is a vertical asymptote because 2 is a zero of the denominator in the reduced form.

there are no common factors no holes

Find all vertical asymptotes and holes of 2

3( )

2 3

xf x

x x

Find all vertical asymptotes and holes of 2

2 14( )

6 7

xf x

x x

, 2

, 2

as x y

and

as x y

y=2 is a horizontal asymptote because___

2 1( )

1

xf x

x

horizontal asymptote

at y = 2

Horizontal Asymptotes

, 0

, 0

as x y

and

as x y

2

1( )

1f x

x

y=0 is a horizontal asymptote because ____

horizontal asymptote

at y = 0

(let n = degree of numerator and d = degree of denominator )

a. If n < d, then y = 0 is the horizontal asymptote

2

5

3 5 1

xy

x x

Horizontal asymptote at y = 0

5 3

2 3

5 1

xy

x x

Horizontal asymptote at y = 0

Examples:

n

n

ay

b

Leading coefficient of numerator

Leading coefficient of denominator

b. If n = d, then is the horizontal asymptote

3 5

2 6

xy

x

Horizontal asymptote at y = 3/2

2

2

10 5 5

5 2 6

x xy

x x

Horizontal asymptote at y = 10/5 = 2

Examples:

c. If n > d, then there is no horizontal asymptote

2 4

2

x xy

x

4 3

2

3 3 4 1

2

x x xy

x

No horizontal asymptotes

No horizontal asymptotes

Identify any vertical or horizontal asymptotes, and any holes in the graph

x 5

1. yx 3

2

x 12. y

x 1

2x x 23. y

x 1

22x x 64. y

2x 3

2

2

3x5. y

x 1

2

x 16. y

x 2x 3

Slant Asymptotes Slant Asymptotes occur when the degree of the occur when the degree of the numerator is exactly one more than the degree of the numerator is exactly one more than the degree of the denominator of the reduced fraction denominator of the reduced fraction .

To find the equation of a slant asymptote use long To find the equation of a slant asymptote use long divisiondivision.

Equation of the horizontal asymptote Equation of the horizontal asymptote is

3 2

2

1

1

x x xy

x x

2

2 1

1

xy x

x x

y x

Ex. Find the equation of the slant asymptote of the equation

3 2

2

1

1

x x xy

x x

Find the equation of the slant asymptote

Equation of the horizontal asymptote Equation of the horizontal asymptote is y = x-2

2

1

x xy

x

Find the equation of the slant asymptote

Equation of the horizontal asymptote Equation of the horizontal asymptote is y = x

2 2

1

x xy

x

State the domain of the function

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

Find and plot the x-intercepts by finding the zeros of the numerator Sketch the vertical asymptotes using dashed vertical lines, and holes using open circles.

Find and sketch any horizontal asymptote using dashed lines.

Find and sketch any slant asymptote using dashed lines.

Plot at least one point between and one point beyond each x-intercept and vertical asymptote.Use smooth curves to complete the graph

The cost c of producing x units of a product is given by

And the average cost per unit is given by

Graph the average cost function, and estimate the number of units that should be produced to minimize the average cost per unit.

2C 0.2x 10x 5

2C 0.2x 10x 5C , x 0

x x

Average Cost

The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is given by

a.Determine the horizontal asymptote of the function and interpret its meaning in the context of the problemb.Graph the function and approximate the time when the bloodstream concentration is the greatest.c.When is the concentration less than 0.345?

2

3

3t tC , t 0

t 50

Medicine.