1. Find the solution(s) to each equation. 1)(x-3)(x+3) = 0 2) (x-4)(x+1) = 0 3) x(x 2 – 1) = 04) x 2 – 4x – 5 = 0 x – 3 = 0 and x +3 = 0 x = 3 and x = -3 x(x – 1)(x + 1) = 0 x – 4 = 0 and x +1 = 0 x = 4 and x = -1 (x – 5)(x + 1) = 0 x – 5 = 0 and x +1 = 0x = 0, x – 1 = 0, and x +1 = 0 x = 0, x = 1, and x = -1 x = 5 and x = -1

2. f(x) = = Has the following characteristics: vertical asymptote is at each real zero of q(x). 9.3a Rational Functions and Their Graphs GRAPHS OF RATIONAL FUNCTIONS Objective – To be able to graph general rational functions. p(x) a x m + …… q(x) b x n + ……

3. Example 1 Find any points of discontinuity (vertical asymptotes) y = 3 x 2 – x – 12 (x – 4)(x + 3) x – 4 = 0 and x + 3 = 0 +4 x = 4 and x = -3 -3

4. On White Board Find any points of discontinuity (vertical asymptotes) y = 1 x 2 – 16 (x – 4)(x + 4) x – 4 = 0 and x + 4 = 0 +4 x = 4 and x = -4 -4

5. On White Board Find any points of discontinuity (vertical asymptotes) y = x + 1 x 2 + 2x – 8 (x + 4)(x – 2) x + 4 = 0 and x – 2 = 0 -4 x = -4 and x = 2 +2

6. On White Board Find any points of discontinuity (vertical asymptotes) y = x 2 – 1 x 2 + 3 x 2 + 3 = 0 x 2 = -3 -3 None

7. f(x) = = Has the following characteristics: GRAPHS OF RATIONAL FUNCTIONS p(x) a x m + …… q(x) b x n + …… horizontal asymptote: - If m < n, the line y = 0 is a horizontal asymptote. - If m = n, the line y = is a horizontal asymptote. - If m > n, the graph has no horizontal asymptote. abab

8. Example 2 Find the horizontal asymptote of: y = – 4x + 3 2x + 1 **Remember that if the exponent is the same you use the coefficients. y = -4 / 2 y = – 2

9. On White Board Find the horizontal asymptote of: y = – 2x + 6 x – 1 **Remember that if the exponent is the same you use the coefficients. y = -2 / 1 y = – 2

10. On White Board Find the horizontal asymptote of: y = 2x + 5 x 2 + 1 y = 0

11. On White Board Find the horizontal asymptote of: y = 4x 5 + 5 x 2 + 1 None

12. f(x) = = Has the following characteristics: 1.x-intercepts are the real zeros of p(x). 2.vertical asymptote is at each real zero of q(x). 3.horizontal asymptote: - If m < n, the line y = 0 is a horizontal asymptote. - If m = n, the line y = is a horizontal asymptote. - If m > n, the graph has no horizontal asymptote. ambnambn 9.3b Rational Functions and Their Graphs GRAPHS OF RATIONAL FUNCTIONS Objective – To be able to graph general rational functions. p(x) a m x m + …… q(x) b n x n + ……

13. Example 1 Graph y = x y -4 -5 0 1 4 5 9.1 61/461/4 4x 2 x 2 – 9 -1 / 2 0 Where are the asymptotes? VA: x = 3, x = -3 HA: y = 4/1 HA: y = 4 –5–4–3–2–112543 –5 –4 –3 –2 –1 1 2 5 4 3 9.1 61/461/4

14. Example 2 Graph the function y = x y 0 1 -3 -4 4 5 -1 / 6 x + 1 (x – 3)(x + 2) -1 / 3 0 –5–4–3–2–112543 –5 –4 –3 –2 –1 1 2 5 4 3 VA:x = 3 and x = -2 -3 / 14 HA:y = 0 5/65/6 6 / 14

15. Example 3 Graph y = x y 1 0 3 4 5 0 -1 / 2 x 2 – 2x +1 x – 2 -4 / 3 4 9/29/2 Where are the asymptotes? x = 2 –5–4–3–2–112543 –5 –4 –3 –2 –1 1 2 5 4 3 16 / 3

16. Sec. 9.4 Multiplying and Dividing Rational Expressions Objective: Multiply/Divide and Simplify Rational Expressions

17. Example 1 Simplify x 2 – 7x – 18 x 2 – 8x -9

18. First Factor top and bottom (x – 9)(x + 2) (x – 9)(x + 1) Then cancel like terms and get (x + 2) (x + 1) -18 -7 -92 Numerator Bottom -9 -8 -9 1

19. Example 2 Simplify

20. See if you can cross cancel anything. 2 2 3 =

21. Example 3 Simplify

22. First do like example 1 and factor everything (x + 6)(x – 2) (x – 7)(x + 5) (x + 6)(x + 5) (x + 4) Cancel all appropriate parts We Get (x – 2)(x – 7) or x 2 – 9x + 14 x + 4x + 4 -12 4 6 -2 1 st Numerator 1 st Denominator 30 11 56 2 nd Numerator -35 -2 -7 5

23. Example 4 Simplify

24. What do we ever do when we divide by a fraction. Yes we multiply by the reciprocal

25. Do the same steps from before and we get (x + 5)(x – 5) (x - 6)(x + 3) (x + 3)(x – 1)x + 5 After Cross Cancelling we end up with (x – 5)(x – 6)orx 2 – 11x + 30 x – 1x - 1

26. Last Example:Complex Fraction Simplify

27. Just like the previous problem we write the bottom fraction by the reciprocal and mult. When you factor you get.

28. An object is 24 cm from a camera lens. The object is in focus on the film when the lens is 12 cm from the film. Find the focal length of the lens. =+ Use the lens equation. 1f1f 1di1di 1do1do =+ Substitute. 1f1f 1 12 1 24 =+Write equivalent fractions with the LCD. 2 24 1 24 == Add and simplify. 3 24 1818 Since =, the focal length of the lens is 8 cm. 1f1f 1818 Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

29. Find the least common multiple of 2x 2 – 8x + 8 and 15x 2 – 60. Step 1: Find the prime factors of each expression. 2x 2 – 8x + 8 = (2)(x 2 – 4x + 4) = (2)(x – 2)(x – 2) 15x 2 – 60 = (15)(x 2 – 4) = (3)(5)(x – 2)(x + 2) Step 2:Write each prime factor the greatest number of times it appears in either expression. Simplify where possible. (2)(3)(5)(x – 2)(x – 2)(x + 2) = 30(x – 2) 2 (x + 2) The least common multiple is 30(x + 2)(x – 2) 2. Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

30. Simplify +. 1 3x 2 + 21x +30 4x 3x + 15 = +Multiply. 1 3(x + 2)(x + 5) 4x(x + 2) 3(x + 2)(x + 5) = Add. 1 + 4x(x + 2) 3(x + 2)(x + 5) = Simplify the numerator. 4x 2 + 8x +1 3(x + 2)(x + 5) = Simplify the denominator. 4x 2 + 8x +1 3x 2 + 21x +30 =+Identity for Multiplication. 1 3(x + 2)(x + 5) 4x 3(x + 5) x + 2 +=+Factor the denominators. 1 3x 2 + 21x +30 4x 3x + 15 1 3(x + 2)(x + 5) 4x 3(x + 5) Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

31. Simplify – 2x x 2 – 2x – 3 = Simplify. 4(2x) – (3)(x – 3) 4(x + 1)(x – 3) = Simplify. 5x + 9 4x 2 – 8x – 12 Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples 3 4x + 4 –=–Factor the denominators. 2x x 2 – 2x – 3 3 4x + 4 2x (x – 3)(x + 1) 3 4(x + 1) = – Identity for Multiplication. x – 3 2x (x – 3)(x + 1) 3 4(x + 1) 4444.

32. Simplify. 1x1x 1y1y 2y2y 1x1x + – 1x1x 1y1y 2y2y 1x1x + – 1x1x 1y1y 2y2y 1x1x + – xy = The LCD is xy. Multiply the numerator and denominator by xy. 2 xy y 1 xy x 1 xy x 1 xy y + – = Use the Distributive Property. y + x 2x – y = Simplify. Method 1:First find the LCD of all the rational expressions. Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

33. (continued) = Write equivalent expressions with common denominators. 1x1x 1y1y 2y2y 1x1x + – + – y xy x xy 2x xy x xy 2x – y xy x + y xy = Add. Method 2: First simplify the numerator and denominator. =÷ Divide the numerator fraction by the denominator fraction. x + y xy 2x – y xy = Multiply by the reciprocal. x + y xy 2x – y = x + y 2x – y Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

34. Solve =. Check each solution. 1 x – 3 6x x 2 – 9 x 2 – 9 = 6x(x – 3)Write the cross products. x 2 – 9 = 6x 2 – 18xDistributive Property –5x 2 + 18x – 9 = 0Write in standard form. 5x 2 – 18x + 9 = 0Multiply each side by –1. (5x – 3)(x – 3) = 0Factor. 5x – 3 = 0 or x – 3 = 0 x = or x = 3Zero-Product Property 3535 1 x – 3 6x x 2 – 9 = Solving Rational Equations LESSON 9-6 Additional Examples

35. (continued) Check:When x = 3, both denominators in the original equation are zero. The original equation is undefined at x = 3. So x = 3 is not a solution. When is substituted for x in the original equation, both sides equal –. 3535 5 12 Solving Rational Equations LESSON 9-6 Additional Examples

36. Solve – =. 3 5x 4 3x 1313 3 5x 4 3x 1313 – =. 9 – 20 = 5xSimplify. 15x– = 15xMultiply each side by the LCD, 15x. 3 5x 4 3x 1313 –= Distributive Property 45x 5x 60x 3x 15x 3 – = x 11 5 Since – makes the original equation true, the solution is x = –. 11 5 11 5 Solving Rational Equations LESSON 9-6 Additional Examples

37. Josefina can row 4 miles upstream in a river in the same time it takes her to row 6 miles downstream. Her rate of rowing in still water is 2 miles per hour. Find the speed of the river current. Relate:speed with the current = speed in still water + speed of the current, speed against the current = speed in still water – speed of the current, time to row 4 miles upstream = time to row 6 miles downstream Write: = 6 (2 + r ) 4 (2 – r ) Solving Rational Equations LESSON 9-6 Additional Examples Define: Distance (mi) Rate (mi/h) Time (h) With current 6 2 + r Against current 4 2 – r 4 (2 – r ) 6 (2 + r )

38. (continued) (2 – r )(6) = (2 + r )(4)Simplify. 4 = 10rSolve for r. 0.4 = rSimplify. The speed of the river current is 0.4 mi/h. = 6 (2 + r ) 4 (2 – r ) (2 + r )(2 – r ) = (2 + r )(2 – r )Multiply by the LCD (2 + r )(2 – r ). 6 (2 + r ) 4 (2 – r ) Solving Rational Equations LESSON 9-6 Additional Examples 12 – 6r = 8 + 4rDistributive Property

39. Find the LCD LCD is (k+1)(k-2) Multiply everything by LCD

40. (continued) Combine Like Terms Combine 2k 2 and k’s to left side