2.7 Graphs of Rational Functions

Students will analyze and sketch graphs of rational functions.

Students will sketch graphs of rational functions that have slant asymptotes.

Students will use rational functions to solve real-life problems.

Guidelines for graphing rational functions by hand:

1. Simplify if possible (by factoring if necessary).

2. Find the Vertical and Horizontal Asymptotes (if any).

3. Evaluate f(0) to find the y-intercept (if any).

4. Set Numerator = 0 to find the zeros of the function (x-intercepts-if any).

5. Plot enough points to determine the sides of the asymptotes that the curve exists.

6. Use smooth curves to complete the sketch.

Example 1: Sketch the graph of a rational function

Sketch the graph of by hand.g xx

( ) 3

2 y

x–2

2

Example 2: Sketch the graph of a rational function

Sketch the graph of by hand.f x xx

( ) 2 1

y

x–2

2

Example 3: Sketch the graph of a rational function

Sketch the graph of .y

x–2

2

f xx

x x( )

2 2

Example 4: Sketch the graph of a rational function

Sketch the graph of .y

x–2

2

f x xx x

( )

2

2

92 3

Slant Asymptotes:

When the degree of the numerator is higher than the degree of the denominator, there are no horizontal asymptotes.

When the degree of the numerator is exactly one higher than the degree of the denominator, there exists a slant asymptote.

To find the slant asymptote, divide the numerator by the denominator (use synthetic or long division). If the constant remainder is disregarded, the result is the slant asymptote.

Example 5: A rational function w/ a slant asymptote

Sketch the graph of .

y

x–2

2

f x x xx

( )

2 21

Example 6: Finding a Minimum AreaA rectangular page is designed to contain 48 square inches of

print. The margins on each side of the page are 1.5 inches wide. The margins at the top and bottom are each 1 inch deep. What should the dimensions of the page be so that the minimum amount of paper is used?