Warm-up Determine the: 1) degree 2) x and y intercepts 3) zeros, their multiplicity, and behavior of the graph at each 4) end behavior Graph it.

3.3 Properties of Rational Functions

1. Rational Functions A rational function is a function of the form where p and q are polynomial functions.

2. Domain and Intercepts 1. Find the domain: (Denominator cannot be zero) 2. Find the y-intercept Find x-intercepts. 4

Domain and Intercepts 1. Find the domain:) 2. Find the y-intercept Find x-intercepts. 5

The graph of a function will never cross a Vertical Asymptote 3. Vertical Asymptotes is the vertical asymptote for f(x) when: as The graph of a function will never cross a Vertical Asymptote 6

4. Finding Vertical Asymptotes of If (c is a zero of the denominator) and If (c is not a zero of numerator) Then is a vertical asymptote of

A hole in a graph A hole occurs at x = c when both: and

More examples Find the vertical asymptotes and holes, if any, of the graphs of the following functions.

5a. Horizontal Asymptotes A horizontal asymptote describes the end behavior of the graph! as The graph of a function may cross a Horizontal Asymptote (usually near the center of the graph). A rational function’s graph will have only one horizontal asymptote. 10

5b. Oblique Asymptotes oblique asymptote When the y-values get close to a line that is not horizontal as the x values approach , then we say the graph has a oblique asymptote y = mx + b If an asymptote is neither vertical or horizontal, it is called an Oblique asymptote An oblique asymptote describes the end behavior of the graph! The graph of a function may cross an OA

Horizontal Asymptotes n = degree of numerator m=degree of denominator Asymptotes Example bottom heavy (denominator is higher degree) n<m y=0 equal (same degree) n=m

Oblique Asymptotes y = quotient in long division Asymptotes Example top heavy (degree of numerator is larger) m > n y = quotient in long division

Horizontal Asymptotes Case 1: Bottom Heavy Case 2: Equal degrees

Horizontal or Oblique Asymptote? State whether the following functions contain a horizontal or oblique asymptote. Then, find it!

Application Problems Ex 1. A population P after t years is modeled by What is the horizontal asymptote for this function? Ex. 2. We wish to enclose an area that is 800 sq. ft. The minimum amount of fencing required to enclose 3 sides can be modeled with Are there vertical or horizontal asymptotes?

Practice

Write an equation for the rational function with the following properties: Vertical Asymptote at x=0, -2, 3. x-intercepts at x=.5, 1, 4. Horizontal Asymptote y=1/3 A hole at x = 2.