2. 1 Warm-up Solve the following rational equation.

3. 2 Set Equation to ZERO Next Slide

4. 3 Problem Continued MUST CHECK ANSWERS x = -4 does not work

5. Discontinuities Section 2-6

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Objectives I can identify Graph Discontinuities –Vertical Asymptotes –Horizontal Asymptotes –Slant Asymptotes –Holes I can find “x” and “y” intercepts

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Rational Functions A rational function is any ratio of two polynomials, where denominator cannot be ZERO! Examples:

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Asymptotes Asymptotes are the boundary lines that a rational function approaches, but never crosses. We draw these as Dashed Lines on our graphs. There are three types of asymptotes: –Vertical –Horizontal (Graph can cross these) –Slant

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Vertical Asymptotes Vertical Asymptotes exist where the denominator would be zero. They are graphed as Vertical Dashed Lines There can be more than one! To find them, set the denominator equal to zero and solve for “x”

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example #1 Find the vertical asymptotes for the following function: Set the denominator equal to zero x – 1 = 0, so x = 1 This graph has a vertical asymptote at x = 1

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 1263457891010 4 3 2 7 5 6 8 9 x- axis y- axis 0 1-2-6 -3-4-5 -7-8-9 1010 -4 -3 -2 -7 -5 -6 -8 -9 0 Vertical Asymptote at X = 1

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Other Examples: Find the vertical asymptotes for the following functions:

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Horizontal Asymptotes Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary. Graph can cross these. To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator See next slide:

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Horizontal Asymptote Given the Rational Function: If m < n, then y = 0 is the HA If m = n, then y = a m /b n is the HA If m > n, then the graph has NO HA

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example #1 Find the horizontal asymptote for the following function: Since the degree of numerator is equal to degree of denominator (m = n) Then HA: y = 1/1 = 1 This graph has a horizontal asymptote at y = 1

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 1263457891010 4 3 2 7 5 6 8 9 x- axis y- axis 0 1-2-6 -3-4-5 -7-8-9 1010 -4 -3 -2 -7 -5 -6 -8 -9 0 Horizontal Asymptote at y = 1

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Discontinuities There may be places where a graph is discontinuous called holes. A hole exists when the same factor exists in both the numerator and denominator of the rational expression. To find the holes, you must fully factor both numerator and denominator and check for like factors.

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Slant Asymptotes (SA) Slant asymptotes exist when the degree of the numerator is one larger than the denominator. Cannot have both a HA and SA To find the SA, divide the Numerator by the Denominator. The results is a line y = mx + b that is the SA.

19. HOLES To Find Holes 1) Factor. 2) Reduce. 3)A hole is formed when a factor is eliminated from the denominator. 4)Set eliminated factor = 0 and solve for x. 5)Find the y-value of the hole by substituting the x-value into reduced form and solve for y. 6)Your answer is written as a point. To Find Holes 1) Factor. 2) Reduce. 3)A hole is formed when a factor is eliminated from the denominator. 4)Set eliminated factor = 0 and solve for x. 5)Find the y-value of the hole by substituting the x-value into reduced form and solve for y. 6)Your answer is written as a point.

20. To find Vertical Asymptote(s) 1) Set reduced denominator = 0 2)Solve for x = #. 3)Your answer is written as a line. To find Vertical Asymptote(s) 1) Set reduced denominator = 0 2)Solve for x = #. 3)Your answer is written as a line.

21. To find x- intercept(s) 1)Set reduced numerator = 0 2) Solve for x. 3) Answer is written as a point. (#, 0) To find x- intercept(s) 1)Set reduced numerator = 0 2) Solve for x. 3) Answer is written as a point. (#, 0)

22. To find y- intercept 1) Substitute 0 in for all x’s in reduced form. 2)Solve for y. 3)Answer is a point. (0, #) To find y- intercept 1) Substitute 0 in for all x’s in reduced form. 2)Solve for y. 3)Answer is a point. (0, #)

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Homework WS 5-3