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

 

Quick Review  

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

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

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

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

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.

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.

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

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.

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

Real World Example

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