Bellwork

168178)( 234 xxxxxf

2. Find all zeros of the function, write the polynomial as a product of linear factors.

1. Find a polynomial function with integer coefficients that has the given zeros.

4, 3i

Last Nights Homework

3637.45

343310.43

2525.41

24

23

23

xx

xxx

xxx

53. -3/2, ±5i55. -3, 5, ±2i65. No, Setting h = 64 and solving the resulting equation yields imaginary roots,

2.6 Rational Functions and Asymptotes-How to find domains of rational functions?-How to find horizontal and vertical asymptotes of graphs of rational functions?

Rational Functions and Asymptotes• A rational function can be written in the form

)(

)()(

xD

xNxf

The most basic rational function

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

Domain: (-∞,∞)

Horizontal Asymptote: x = 0

Vertical Asymptote: y = 0

• The line x = a is a vertical asymptote of the graph of f if f(x)→∞ or f(x)→-∞ as x→a, either from the right or the left.

• The line y = b is a horizontal asymptote of the graph of f if f(x)→b as x→∞ or x→-∞.

Vertical Asymptotes

• Let f be the rational function

)(

)()(

xD

xNxf

Where N(x) and D(x) have no common factors.

The graph of f has vertical asymptotes at the zeros of D(x).

Example 1: Find the Vertical Asymptotes.

2

4)()

2

x

xxfa

xx

xxgb

3

3)()

2

209

82)()

2

xx

xxwc

Hole atx = -2No VA

VA at x = 0, Hole at x = 3

VA at x = 5

Hole at x = 4

Horizontal Asymptotes

• The graph f has at most one horizontal asymptote determined by looking at the exponents of the numerator and the denominator. • If n < m, then y = 0 is the H.A.• If n = m, then y = a/b is the H.A.• If n > m, then there is no H.A.

...

...)(

m

n

bx

axxf

Example 2: Find the H.A. of the following functions.

13

2)()

2 x

xxfa

13

2)()

2

2

x

xxgb 13

2)()

2

3

x

xxhc

Bigger exponent in D(x).H.A. y = 0

Same exponent in N(x)and D(x). H.A. y = 2/3

Bigger exponent inN(x). No H.A.

Example 3: Find a functions vertical asymptotes,

and horizontal asymptotes.

54

273)()

3

23

x

xxxfa

3

4

5.. xAV

4

3. yAH

12

42)()

2

23

x

xxxgb

4

1,4

1 xxVA

0yHA

Tonight’s Homework• Pg 195 #7-19. #40, 41