2.7 – Graphs of Rational Functions

By then end of today you will learn about…….

Rational Functions

Transformations of the Reciprocal function

Limits and Asymptotes

Analyzing Graphs of Rational Functions

Rational FunctionsRational functions are ratios (or quotients) of

polynomial functions

Definition: Let f and g be polynomial functions with g(x) ≠ 0. Then the function given by

y(x) = f(x)

g(x)

is a rational function

Domain of a Rational Function

f(x) = 1

x+ 3

Vertical Asymptote:

“As x approaches -3 from the left, the values of f(x) decrease infinitely.”

Domain:

Use limits to describe its behavior at the vertical asymptotes:

“As x approaches -3 from the right, the values of f(x) increase infinitely.”

The Basic Reciprocal Function

f(x) = 1

x•Domain:•Range:•Continuity:•Decreasing on:•Symmetry:•Bounded?•Extrema?•Horizontal Asymptote: •Vertical Asymptote:•End behavior

Transforming the Reciprocal Function

Use your calculators to graph the following functions. How do they compare to the basic reciprocal function?

f(x) = -3

x-1

Transforming the Reciprocal Function

•Transformations:

•Domain:

•Horizontal Asymptote:

•Vertical Asymptote:

•Behavior of f(x) at value of x not in domain (or vertical asymptotes):

Graphs of Rational Functions

End Behavior Asymptote: If numerator degree < denominator degree, then the end

behavior asymptote is the horizontal asymptote y=0 If numerator degree = denominator degree, then the end

behavior asymptote is the horizontal asymptote:y= leading coefficient of numerator

leading coefficient of denominator

If numerator degree > denominator degree, the end behavior asymptote is the quotient polynomial function y=q(x), where f(x) = g(x)q(x) + r(x)

There is NO horizontal asymptote.

Vertical Asymptotes: Occur at the zeros of the denominator

X-Intercepts: Occur at the zeros of the numerator

Y-intercept: This is the value of f(0), if defined

Transforming the Reciprocal Graph

f(x) = 2x -1

x + 3

•Transformation:

•Vertical Asymptote:

•Horizontal Asymptote:

•Limits to describe f(x) at vertical asymptote:

h(x) = x – 1 x2 – x – 12

•X-Intercept:•Y-Intercept:•Vertical Asymptotes:•Horizontal Asymptote:•Behavior at Vertical Asymptotes:

•Domain:•Range:•Continuity:•Increasing•Decreasing:•Symmetry?•Extrema?•End behavior:

Find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the rational function

•X-Intercept:•Y-Intercept:•Vertical Asymptotes•Horizontal Asymptote•Behavior at Vertical Asymptotes

•Domain•Range•Continuity•Decreasing•Symmetry?•Extrema?•End behavior

f(x) = x2 + x -2 x2 - 9

h(x) = x3 - 2

x + 2•X-Intercept•Y-Intercept•Vertical Asymptotes•End behavior Asymptote•Behavior at Vertical Asymptotes

•Domain•Range•Continuity•Decreasing•Increasing:•Symmetry?•Extrema?•End behavior

Find the intercepts, analyze, and draw the graph of the rational function

Slant Asymptote

f (x)=x3

x2 −9• End Behavior Asymptote:

•Vertical Asymptote:

•X-Intercept:

•Y-Intercept:

Don’t forget your homework!

pg. 246-247 (4-60, every 4)