1. Alg 2 Warm Up – Wed (5/15)-Thurs (5/16) 1.List the possible roots. Then find all the zeros of the polynomial function. f(x) = x 4 – 2x 2 – 16x -15 Answers: Possibles: +/- 1, 3, 5, 15 {3, -1,-1+2i,-1-2i}

2. Announcements Section 3.5 – Rational Functions and their Graphs Objective: Find the domain of rational functions. Use arrow notation. Find the vertical and horizontal asymptotes. Graph Rational Functions

3. 3.5 – Finding the domain of a Rational Function To find the domain set the denominator to zero and solve for x. The domain will be all real number except that value for x. The denominator is x- 3. Solve for x. x- 3 = 0 x = 3 The domain is all Real Numbers Except x≠3

4. Example 2 Find the domain: X 2 + 9 = 0 X 2 = -9 The domain is only for real numbers not imaginary the domain is(-∞, ∞)

5. YOU TRY!!! Find the domain of Answers 1. All real numbers except x≠5 or -5 2. All real numbers.

6. Reciprocal Function The reciprocal function is the most basic rational function. It is f(x) = 1/x. It looks like: If x is far from zero, then 1/x is close to zero. If x is close to zero, then 1/x is far from zero.

7. Arrow Notation

8. How to use Arrow Notation If you have x 3 - means what is the graph doing as x approaches 3 from the left. Left because of the negative attached.

9. Example Let’s look at problem 9 It is asking to look as x approaches -3 from the left what is the graph doing? Look at the graph from the left and the first curve is going to -∞ so the answer is f(x) -∞. You Try #13 Answer: f(x) is approaching zero.

10. The parent function of rational functions is What does the graph look like?

11. Another Basic Rational Function f(x) = 1/x 2 and it looks like this:

12. Asymptotes: lines that a graph approaches but does not cross Vertical asymptotes:  Whichever values are not allowed in the domain will be vertical asymptotes on the graph.  Where is the domain limited?  Set those factors that only appear in the denominator or those that appear more times in the denominator than numerator equal to zero and solve. denominator

13. Definition of Vertical Asymptote

14. Example 1.Find the vertical asymptotes: Set x – 2 = 0, x = 2 is a vertical asymptote. 2.Find the vertical asymptotes: Factor x 2 – 9 = (x-3)(x+3) Set x – 3 = 0, x =3 is the vertical asymptote There won’t be one at x=-3, which means there is a hole in the graph at -3 or point discontinuity. Picture next slide.

15. Picture If the factors are not real, then we say no vertical asymptotes.

16. Horizontal asymptotes: Look at the degrees of the numerator and denominator  If the degrees are equal then the horizontal asymptote is the ratio of the leading coefficients ( y = ratio of leading coefficients)  If the degree in the denominator is greater then the horizontal asymptote is y = 0  If the degree in the numerator is greater then there is no horizontal asymptote.

17. Definition of Horizontal Asymptotes

18. Ex 1: Find the horizontal asymptotes of each rational function A)  Horizontal: degrees are equal (both are 1 st degree) so y = ratio 3/1 y = 3 B) Horizontal: If the degree in the denominator is greater than the numerator horizontal asymptote is y = 0 C) If the degree in the numerator is greater than there is no horizontal asymptote.

19. Pictures of A,B, and C

20. YOU TRY!!! Ex 1: find the vertical & horizontal asymptotes of each rational function A) Vertical: x-5=0 x = 5  Horizontal: degrees are equal (both are 1 st degree) so y = ratio 4/1 y = 4 B) Vertical: x+1 = 0 x+3=0 So x = -1, x = -3, Horizontal: denominator is bigger (3 rd degree vs. 2 nd ) so y = 0 x = -2 is a hole

21. Lesson 3.5 Graphing a Rational Function Rational Functions that are not transformations of f(x) = 1/x or f(x) = 1/x 2 can be graphed using the following suggestions. Strategy for Graphing a Rational Function The following strategy can be used to graph f(x) = p(x) q(x), Where p and q are polynomial functions with no common factors.

22. Seven steps: Step 1: Determine whether the graph of f has symmetry. f(-x) = f(x): y-axis symmetry f(-x) = -f(x): origin symmetry Step 2: Find the y-intercept (if there is one) by evaluating f(0). Step 3: Find the x-intercepts (if there are any) by solving the equation p(x) = 0.

23. Steps continued: Step 4: Find any vertical asymptote(s) by solving the equation q(x) =0. Step 5: Find the horizontal asymptote (if there is one ) using the rule for determining the horizontal asymptote of a rational function. Step 6: Plot at least one point between and beyond each x-intercept and vertical asymptote. Step 7: Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

24. Example 1: Graph: f(x) = 2x x-1

28. Example 2: Step 1: Determine Symmetry: f(-x) Step 2: Find the Y-intercept. f(0) Step 3: Find the x-intercepts. p(x) = 0 Step 4: Find the vertical asymptote(s). q(x)=0 Step 5: Find the horizontal asymptote. (Degree of numerator and denominator) Step 6: Plot points between and beyond each x-intercept and vertical asymptote. (table) Step 7: Graph the function.

29. The graph of f is symmetric with respect to the y-axis. 2.f(0) = 3*0 2 = 0 = 0 0 2 – 4 -4 The y-intercept is 0, so the graph passes thru the origin. 3. 3x 2 =0, so x = 0. The x-intercepts is 0, verifying the graph passes through the origin. 4.Set q(x) =0, x 2 -4 = 0, so x = 2 and x= -2 The vertical asymptotes are x = -2 and x=2.

30. Ex. 2 cont: 5.Look at the degree of numerator and denominator. They are equal so you use the leading coefficients. 3/1 The horizontal asymptote is y = 3. 6. Plot points between and beyond each x-intercept and vertical asymptote. X-3134 f(x) 3x 2 x 2 -4 27/ 5 27/54

31. 7. Graph the functions.

32. Summary: If you are given the equation of a rational function, explain how to find the vertical asymptotes of the function.