8.2 Graph Simple Rational Functions

Rational function

A function of the form where p(x) and g(x) are polynomials and g(x) ¹ 0

PARENT FUNCTION FOR SIMPLE RATIONAL FUNCTIONS

• The graph of the parent function f(x) = is a hyperbola , which consists of two symmetrical parts called branches .
• The domain and range are all nonzero real numbers .
• The asymptotes are x = _0_, and y = _0_ .
• Any function of the form g(x) = (a = 0) has the same _asymptotes_ , _domain_, and _range_ as the function f(x) = .

Example 1

Graph a rational function of the form y =

Graph the function y =.

1. Draw the asymptotes x = __0__and y = __0__ .

2. Plot points to the left and to the right of the vertical asymptote, such as
(-2, _-2_ ), (-1, _- 4_ ), (1, _4_ ), and (2, _2 ).

3. Draw the branches of the hyperbola so that they pass through the plotted points and approach the asymptotes.

GRAPHING TRANSLATIONS OF SIMPLE RATIONAL FUNCTIONS

To graph a rational function of the form y = + k, follow these steps:

Step 1 Draw the asymptotes x = _h_ and y =__k__.

Step 2 Plot points to the left and to the right of the vertical asymptote .

Step 3 Draw the two branches of the hyperbola so that they pass through the plotted points and approach the asymptotes.

Example 2

Graph y = . State the domain and range.

1. Draw the asymptotes x = -3 and y = _2_ .

2. Plot points to the left and to the right of the vertical asymptote, such as (-6, 4 ),
(-4, _8_ ), (-2, -4_ ), and (0, _0_ ).

3. Draw the two branches of the hyperbola so that they pass through the plotted points and approach the asymptotes.

The domain is _all real numbers except -3_ , and the range is _all real numbers except 2_

Example 3

Graph a rational function of the form

Graph State the domain and range.

Solution

1. Draw the asymptotes. Solve x - 1 = 0 for x to find the vertical asymptote x = _1_. The horizontal asymptote is the line

y = = = 4

2. Plot points to the left of the vertical asymptote, such as (-1, _3_ ) and (0, _2_), and points to the right, such as (2, _6_ ) and (3, _5_ ).

3. Draw the two branches of the hyperbola so that they pass through the plotted points and approach the asymptotes.

The domain is _all real numbers except 1_ . The range is _all real numbers except 4_ .

domain: all real numbers except -2

range: all real numbers except 2