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1 page for notes, 1 page for

homework

(6 graphs on homework)

Section 8-3 1

Bell Quiz 8-3

2Section 8-3

10 pts

possible

4 pts

6 pts 1. 2x– .Graph y =

2. Find the vertical and horizontal asymptotes of y = .4x + 5

x – 1

8-3 Chapter 8: Rational Functions

3Section 8-3

8-3

4Section 8-3

Chapter 8: Rational Functions

Key Concept

Graphs of Rational FunctionsGraphs of Rational FunctionsGraphs of Rational FunctionsGraphs of Rational FunctionsLet p(x) and q(x) be polynomials with no common factors other than ±1. The graph of the following rational function has the characteristics listed below.

� � �����

�����

������

��⋯�����

���������

���⋯�����

1.1.1.1. The The The The xxxx----intercepts of the graph of intercepts of the graph of intercepts of the graph of intercepts of the graph of ffff are the real zeros of are the real zeros of are the real zeros of are the real zeros of pppp((((xxxx).).).).

2.2.2.2. The graph of The graph of The graph of The graph of ffff has a vertical asymptote at each real zero of has a vertical asymptote at each real zero of has a vertical asymptote at each real zero of has a vertical asymptote at each real zero of qqqq((((xxxx).).).).

3.3.3.3. The graph of The graph of The graph of The graph of ffff has at most one horizontal asymptote, which is determined has at most one horizontal asymptote, which is determined has at most one horizontal asymptote, which is determined has at most one horizontal asymptote, which is determined by the degrees by the degrees by the degrees by the degrees mmmm and and and and nnnn of of of of pppp((((xxxx) and ) and ) and ) and qqqq((((xxxx).).).).

5Section 8-3

� � �

� � �

� � �

The line y = 0 is a horizontal asymptote.

The line y = ��

��is a horizontal asymptote.

The graph has no horizontal asymptote.

The graphs end behavior is the same as the graph of � ���

������

The graph passes through the points (–3, 0.6), (–1, 3),

(0, 6), (1, 3), and (3, 0.6). The domain is all real numbers, and the range is 0 < y ≤ 6.

EXAMPLE 1 Graph a rational function (m < n)

Graph y = . State the domain and range.6

x2 + 1

SOLUTION

The degree of the numerator, 0, is less than the degree of the denominator, 2. So, the line y = 0 (the x-

axis) is a horizontal asymptote.

The numerator has no zeros, so there is no x-intercept. The

denominator has no real zeros, so there is no vertical asymptote.

6

EXAMPLE 2 Graph a rational function (m = n)

Graph y = . 2x2

x2– 9

Section 8-3 7

EXAMPLE 2 Graph a rational function (m = n)

Section 8-3 8

EXAMPLE 3 Graph a rational function (m > n)

Graph y = . x2 +3x – 4

x – 2

Section 8-3 9

EXAMPLE 3 Graph a rational function (m > n)

Section 8-3 10

EXAMPLE 4 Solve a multi-step problem

SKIPSKIPSKIPSKIP

Section 8-3 11

GUIDED PRACTICE for Example 1, 2 and 3

Graph the function.

1. y =4

x2 + 23x2

x2– 12. y =

Section 8-3 12

GUIDED PRACTICE for Example 1, 2 and 3

Graph the function.

3. f (x) =x2 –5x2 +1 4. y =

x2 – 2x – 3x – 4

Section 8-3 13

HOMEWORK

Sec 8-3 (pg 568)

3-5 ALL, 7, 8, 10, 11, 14, 15-22 DO 2 SKIP 1, 37-39 ALL

(SKIP 17, 20)

14Section 8-3