Rational Functions and Asymptotes (Section 2-6)

(a) Find the domain of the function, (b) complete each table, and (c) discuss the behavior of f near any excluded values. Example 1 x f(x) -0.5 -0.1 -0.001 -0.0001 x f(x) 0.5 0.1 0.001 0.0001

(a) Find the domain of the function, (b) complete each table, and (c) discuss the behavior of f near any excluded values. Example 2 x f(x) 0.5 0.9 0.99 0.999 x f(x) 1.5 1.1 1.01 1.001

Parent Function pg 147

Notation: f(x) → - ∞ as x → 0- means f(x) approaches -∞ without bound (i.e. goes towards -∞ as x approaches 0 from the left f(x) → ∞ as x → 0+ means f(x) approaches ∞ without bound (i.e. goes towards ∞ )as x approaches 0 from the right. f(x) → 0 as x → -∞ means f(x) approaches 0 as x approaches - ∞ without bound (i.e. goes towards -∞) f(x) → 0 as x → ∞ means f(x) approaches 0 as x approaches ∞ without bound (i.e. goes towards ∞)

Asymptotes Defined: The line x=a is a vertical asymptote of the graph of f if f(x) → ∞ or f(x) → ∞ as x → a from either the left or the right. The line y = b is a horizontal asymptote of the graph of f if f(x) → b as x → ∞ or x → - ∞ An asymptote is a line (either horizontal or vertical ) that the graph approaches but never touches.

Determining Horizontal and Vertical Asymptotes Vertical: Set the denominator equal to zero and solve Horizontal: Compare Degree of Numerator to Degree of Denominator Horizontal Asymptote Less than y = 0 Equal Greater than None LC = Leading Coefficient Holes: What cancels in denominator

pg 147

Identify any horizontal and vertical asymptotes. Example 3

Identify any horizontal and vertical asymptotes. Example 4

Identify any horizontal and vertical asymptotes. Example 5

Match the function with its graph. Example 6 A B C

(a) Identify any horizontal and vertical asymptotes and (b)identify any holes in the graph. Example 7

(a) Identify any horizontal and vertical asymptotes and (b)identify any holes in the graph. Example 8

(a) Find the domain of the function (b) decide if the function is continuous (c)identify any horizontal and vertical asymptotes. Example 9

(a) Find the domain of the function (b) decide if the function is continuous (c)identify any horizontal and vertical asymptotes. Example 10

(a) Find the domain of the function (b) decide if the function is continuous (c)identify any horizontal and vertical asymptotes. Example 11

HW #61 pg 152-153 (1, 3, 7-12all, 13-21 odd)

Find the zeros (if any) of the rational function. Example 12

Find the zeros (if any) of the rational function. Example 13

Find the zeros (if any) of the rational function. Example 14

Find the zeros (if any) of the rational function. Example 15

If you are given characteristics of a function and asked to write a function that would match those characteristics, think about what each piece means for the numerator and the denominator and then put all of the information together.

Write a rational function f that has the specified characteristics. Example 16 Vertical asymptote: x = -1 Horizontal asymptote: y = 0 Zeros: x = 2

Write a rational function f that has the specified characteristics. Example 17 Vertical asymptotes: x = -1, x = 2 Horizontal asymptote: y = -2 Zeros: x = -2, x = 3

HW #62 pg 153 (23, 25, 31-34 all)