Graphs and Rational Functions

Precalculus

Functions and Graphs

Chapter 2Section 6

Graphs and Rational Functions

Essential Question: What are the key characteristics of a rational function?

Student Objectives:The student will identify the vertical and/or horizontal asymptotes.

The student will identify the end behavior of the function.

The student will identify the translations of the reciprocal functions.

Terms:

End behavior

Horizontal asymptote

Oblique asymptote

Slant asymptote

Vertical asymptote

x-intercept

y-intercept

Graphing Calculator Skills:

The student is expected to graph the rational function in an appropriate window to

illustrate the key concepts of the function.

Theorems and Definitions

Rational Function

Let and be polynomials functions with Then the function given by

is a rational function.

Graph of a Rational Function

The graph of

has the following characteristics:

1. End behavior asymptotes:

If the end behavior is the horizontal asymptote of

If the end behavior asymptote is the quotient

polynomial of where

There is no horizontal

asymptote. There is a slant asymptote if .

2. Vertical asymptotes:

These occur at the real zeros of the denominator, provided

that the zeros are not also the zeros of the

numerator of equal or greater multiplicity.

3. x-intercepts:

These occur at the real zeros of the numerator, which are

also not zeros of the denominator.

4. y-intercepts

This occurs at the value of , if it is defined at this

point.

Sample Questions:

1.Determine the graph of the rational function and determine the following questions concerning the function: .

a. Domain: a. ______

b. Range: b. ______

c. Continuous interval: c. ______

d. Increasing interval: d. ______

e. Decreasing intervals: e. ______

f. Symmetry: f. ______

g. Bounded: g. ______

h. x-intercept(s) h. ______

i. y-intercept: i. ______

j. Relative minimums: j. ______

k. Relative maximums: k. ______

l. Removable points l. ______

m. Horizontal Asymptotes: m.______

n. Vertical asymptotes: n. ______

o. Slant asymptote: o. ______

p. End behavior: p.

2.Determine the graph of the rational function and determine the following questions concerning the function: .