1. Observer Professor Susan Steege
Lesson Plan
MAE4945 Lesson Plan #3
Graphing Rational Functions (Cont.)
Joseph Torres
April 9, 2015
Cooperating Teacher Chris Bayus
Observer Professor Susan Steege

3. Quick Review

4. The rational function f(x) = can be transformed by using methods similar to those used to transform other types of functions. 1 x

5. Like logarithmic and exponential functions, rational functions may have asymptotes. The function f(x) = has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. 1 x

8. Identify the zeros and asymptotes of the function. Then graph.
Example 4C: Graphing Rational Functions with Vertical and Horizontal Asymptotes
Identify the zeros and asymptotes of the function. Then graph.
4x – 12
x – 1
f(x) =
4(x – 3)
x – 1
f(x) =
Factor the numerator.
The numerator is 0 when x = 3.
Zero: 3
Vertical asymptote: x = 1
The denominator is 0 when x = 1.
The horizontal asymptote is y = = = 4. 4 1
leading coefficient of p
leading coefficient of q
Horizontal asymptote: y = 4

10. Identify the zeros and asymptotes of the function. Then graph.
Check It Out! Example 4c
Identify the zeros and asymptotes of the function. Then graph.
3x2 + x
x2 – 9
f(x) =
x(3x – 1)
(x – 3) (x + 3)
f(x) =
Factor the numerator and the denominator.
Zeros: 0 and – 1 3
The numerator is 0 when x = 0 or x = – . 1 3
Vertical asymptote: x = –3, x = 3
The denominator is 0 when x = ±3.
The horizontal asymptote is y = = = 3. 3 1
leading coefficient of p
leading coefficient of q
Horizontal asymptote: y = 3

11. In some cases, both the numerator and the denominator of a rational function will equal 0 for a particular value of x. As a result, the function will be undefined at this x-value. If this is the case, the graph of the function may have a hole. A hole is an omitted point in a graph.

12. Example 5: Graphing Rational Functions with Holes
Identify holes in the graph of f(x) = Then graph.
x2 – 9
x – 3
(x – 3)(x + 3)
x – 3
f(x) =
Factor the numerator.
There is a hole in the graph at x = 3.
The expression x – 3 is a factor of both the numerator and the denominator.
(x – 3)(x + 3)
(x – 3)
For x ≠ 3, f(x) = = x + 3
Divide out common factors.

13. Example 5 Continued
The graph of f is the same as the graph of y = x + 3, except for the hole at x = 3. On the graph, indicate the hole with an open circle. The domain of f is {x|x ≠ 3}.
Hole at x = 3

14. Check It Out! Example 5
Identify holes in the graph of f(x) = Then graph.
x2 + x – 6
x – 2
(x – 2)(x + 3)
x – 2
f(x) =
Factor the numerator.
There is a hole in the graph at x = 2.
The expression x – 2 is a factor of both the numerator and the denominator.
For x ≠ 2, f(x) = = x + 3
(x – 2)(x + 3)
(x – 2)
Divide out common factors.

15. Check It Out! Example 5 Continued
The graph of f is the same as the graph of y = x + 3, except for the hole at x = 2. On the graph, indicate the hole with an open circle. The domain of f is {x|x ≠ 2}.
Hole at x = 2

16. Real World Example

17. -Horizontal Asymptote -Holes -Graph -Domain
Assignment:
-Vertical Asymptote
-Horizontal Asymptote
-Holes
-Graph
-Domain