Class Work

1. Find the real zeros by factoring. P(x) = x4 – 2x3 – 8x + 16

2. Divide.

3. Find all the zeros of the polynomial. P(x) = x3 – 2x2 + 2x – 1

3 24 2 2 3

2 1

x x x

x

Sec 3.6 Rational Functions

Objectives:•To understand how to find holes, vertical, horizontal and slant asymptotes.

Vertical Asymptotes

The line x = a is a vertical asymptote if y approaches ± as x approaches a from the left or right.

Vertical Asymptotes - A rational function has a vertical asymptote at x = c when

• c is a zero of the denominator• c is NOT a zero of the numerator.

Ex 1. Find the vertical asymptotes.2

)5

6)

2 9

ax

bx

Holes

When a number c is both a zero of the numerator and the denominator then there is a hole at x = c.

Ex 2. Find the holes of the function.2

3

4)

2

64)

4

xa

x

xb

x

Horizontal Asymptotes

Let ...

( )...

n

m

axf x

cx

• If n < m, then there is a horizontal asymptote at y = 0.

• If n = m, then there is a horizontal asymptote at a

yc

where a and c are the leading coefficients of the numerator and the denominator.

• If n >m, then there is no horizontal asymptote, but it does have a slant asymptote.

Ex 3. Find all holes and vertical and horizontal asymptotes.a)

b)

c)

2

3 2

2( )

5 6

x xf x

x x x

2

2

3 4( )

2 4

x xf x

x x

2

3 2

3 12( )

20

x xf x

x x x

Slant Asymptotes

Remember, slant asymptotes occur when the degree of the numerator is greater than the degree of the denominator.

To find slant asymptotes, divide the numerator by the denominator and the quotient will give you the equation of the slant asymptote.

Ex 4. Find all asymptotes and holes.

a)

b)

2 4 5( )

3

x xf x

x

2 6( )

2

x xf x

x

This is the graph of Ex 4 part a.

Class Work

Find all asymptotes and holes.4.

5.

6.

2

2

3 4( )

2 4

x xf x

x x

23 2

2 2

2 4 32 5 6( )

3 2 3 2

x x xx x xf x

x x x x

3 2

2

4 4( )

6

x x xf x

x x

HW 3.6 Worksheet