1. Unit 7 –Rational Functions Graphing Rational Functions

2. What to do first FACTOR!!!! – Factor either numerator, denominator, or both, before graphing. – Do NOT simplify/cancel anything… yet.

3. Graphing Rational Functions To sketch these graphs, you must first identify…

4. M4: Vertical Asymptotes Values of x that make the denominator 0. Ex: After factoring we have: Denominator is 0 at x = 4 & x = -1. Those would be vertical asymptotes (graph cannot cross those lines).

5. M4: Zeros Values of x that make the numerator 0. Ex: After factoring we have: Numerator is 0 at x = -3 & x = 2. Those points would be zeros (graph hits x-axis at those points).

6. M4: Holes Values of x that make both numerator & denominator 0. Ex: After factoring we have: Numerator and denominator are 0 at x = -2. That point is a hole in the graph (graph passes through that point, but the function is undefined at that point).

7. M4: Holes Holes are NOT zeros. They are not necessarily on the x-axis. – To find the coordinates of a hole, cancel the common binomial, and plug the value of x into what’s left to find the y value. After simplifying we have: Plugging -2 for x gives: A hole would be located at the point (-2, -4).

8. M4: Horizontal Asymptote Determined by degrees of numerator and denominator. – If numerator degree > denominator degree, no horizontal asymptote. – Ex. Numerator degree = 2, denominator degree = 1. No horizontal asymptote.

9. M4: Horizontal Asymptote Determined by degrees of numerator and denominator. – If numerator degree < denominator degree, there is a horizontal asymptote at y = 0. – Ex. Numerator degree = 1, denominator degree = 2. Horizontal asymptote at y = 0.

10. M4: Horizontal Asymptote Determined by degrees of numerator and denominator. – If numerator degree = denominator degree, the horizontal asymptote is at y = ratio of leading coefficients. – Ex. Degrees are both 2. Ratio of leading coefficients = 3/1. Horizontal asymptote at y = 3.

11. Identifying the Mathtastic 4 After finding asymptotes, zeros, and holes, graphs of rational functions are easy to sketch. – Be sure to use your graphing calculator to check your work.

12. Identifying the Mathtastic 4 Practice identifying the Mathtastic 4 with the functions presented in this presentation. – Keep in mind that all 4 will not always show up in a single function.

13. Homework Textbook Section 8-4 (pg. 598): 33-42 Should be completed before Unit 7 Exam