1. 5.3 Graphs of Rational Functions
MAT SPRING 2009
5.3 Graphs of Rational Functions
In this section, we will study the following topics:
Identifying ‘holes’ in the graph of rational functions
Analyzing and sketching graphs of rational functions

2. Rational Functions That are Not in Lowest Terms: A “Hole” lot of Fun!
MAT SPRING 2009
Rational Functions That are Not in Lowest Terms: A “Hole” lot of Fun!
Example
Consider the function
We know that the domain cannot include ________.
We’ll start by writing the expression in lowest terms.

3. A “HOLE” appears in the graph at this point.
MAT SPRING 2009
Example (cont)
Therefore, the graph of the function will be the same as the graph of ______________ (a straight line), with the EXCEPTION of the point with x-coordinate ____.
A “HOLE” appears in the graph at this point.

4. Finding the y-coordinate of the hole
MAT SPRING 2009
Finding the y-coordinate of the hole
To determine the y-coordinate of the “hole” (point of discontinuity):
SUBSTITUTE THE X-VALUE AT WHICH THE HOLE OCCURS INTO THE REDUCED FORM OF THE RATIONAL EXPRESSION.
The hole occurs at x = -3, so we will substitute this value into the reduced expression:
 A hole appears in the graph at the point __________

5. MAT SPRING 2009
Now…A Look at the Graph
The graph of will look just like the graph of the line
with a hole at (-3, -6).
hole at (-3, -6)

6. And Now…A Look at the Graph Using the Calculator
MAT SPRING 2009
And Now…A Look at the Graph Using the Calculator
The graphing calculator does not show “holes” in the graph well. However, we can use the Value feature or the Table to support the existence of a point of discontinuity.

7. Guidelines for Graphing Rational Functions
Write the rational expression in simplest form.
(Factor the numerator and denominator and divide out any common factors.)
Find the coordinates of any “holes” in the graph. (The x-coordinate comes from the common factor; the y-coordinate is found by substituting the x-coordinate into the reduced form of the function.)
Find and plot the y-intercept, if any, by evaluating f(0). (Substitute x=0 into expression.)
Find and plot the x-intercept(s), if any, by finding the zeros of the numerator. (Set the numerator of the reduced expression equal to zero and solve for x.)

8. Guidelines for Graphing Rational Functions (continued)
Find the vertical asymptote(s), if any, by finding the zeros of the denominator (of the reduced form of the function). Sketch these using dashed lines.
Find the horizontal asymptote, if any, by comparing the degrees of the numerator and denominator. Sketch these using dashed lines.
Find the oblique asymptote, if any, by dividing the numerator by the denominator using long division. Write in the form of y = mx + b, where mx + b is the quotient. Ignore the remainder.
Plot 5-10 additional points, including points close to each x-intercept and vertical asymptote.
Use smooth curves to complete the graph.

9. Sketching the Graph of a Rational Function by Hand
Example
Given ,
State the domain of f in interval form.
State the equations of any vertical and horizontal asymptotes.
State any x-intercepts of the graph of f.
State the y-intercept of the graph of f, if it exists.
Sketch the graph of the function using the information in parts a – d and by plotting several additional points.

10. Sketching the Graph of a Rational Function by Hand
Example (cont)

11. Sketching the Graph of a Rational Function by Hand
Example
Given ,
Write the expression in simplest form.
State the domain of f in interval form.
State the coordinates of any “holes” in the graph of f.
State the equations of any vertical and horizontal asymptotes.
State any x-intercepts of the graph of f.
State the y-intercept of the graph of f, if it exists.
Sketch the graph of the function using the information in parts a – f and by plotting several additional points.

12. Sketching the Graph of a Rational Function by Hand
Example (cont)

13. Sketching the Graph of a Rational Function by Hand
Example
Given ,
State the domain of f in interval form.
State the equations of any vertical, horizontal, and oblique asymptotes.
State any x-intercepts of the graph of f.
State the y-intercept of the graph of f, if it exists.
Sketch the graph of the function using the information in parts a – d and by plotting several additional points.

14. Sketching the Graph of a Rational Function by Hand
Example (cont)

15. Sketching the Graph of a Rational Function by Hand
One More Example
Given ,
State the domain of f in interval form.
State the equations of any vertical, horizontal, and oblique asymptotes.
State any x-intercepts of the graph of f.
State the y-intercept of the graph of f, if it exists.
Sketch the graph of the function using the information in parts a – d and by plotting several additional points.

16. Sketching the Graph of a Rational Function by Hand
One More Example (cont)

17. “You don’t know a hole in your function from your asymptote!”
MAT SPRING 2009
End of Section 5.3
Trash Talking in Math:
“You don’t know a hole in your function from your asymptote!”