1. What’s up with the Education is Power! Objective: Prerequisite Review Date: 9/9/14 Bell Ringer: What is Calculus and why study it? Http://www.youtube.com/watch?v=ismnD_QHKkQ Homework Requests: Rational expressions - Check solutions Absolute Values Memory Sheet In class: Rational Expressions (Tricks) Complex Fractions Homework: Finish Worksheets Rational Expressions (Tricks), Complete Complex Fractions WS Syllabus online Parent sign in sheet due Tues. 9/9 If needed, bring in calculator loan contract 9/9 (online) Bring Calculator Register on morganparkcps.org due Tues. 9/9 Bring in $15.00 fee by FRIDAY Announcements: Monday Day for Mandatory Session Periodic Functions (Unit Circle) Domain and Range Get Email Addresses Find the domain of each function. Use interval and set builder notation 1. 2.

3. Objective Simplify complex fractions Lets Review fraction rules first………….. Complex fractions

5. Simplifying Complex Fractions A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator. Example:

6. Adding/Subtracting Fractions 7 12 = 5 2 + Add. 5 12 2 +

7. Common Denominators 1.Add or subtract the numerators. 2.Place the sum or difference of the numerators found in step 1 over the common denominator. 3.Simplify the fraction if possible. Subtract

8. Common Denominators a.)Add Example:

9. Common Denominators b.) Subtract Example:

10. Unlike Denominators 1.Determine the LCD. 2.Rewrite each fraction as an equivalent fraction with the LCD. 3.Add or subtract the numerators while maintaining the LCD. 4.When possible, factor the remaining numerator and simplify the fraction.

11. Unlike Denominators a.) The LCD is w(w+2). Example:

12. Unlike Denominators b.) The LCD is 12x(x – 1). This cannot be factored any further. Example:

13. Simplifying Complex Fractions A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator. Example:

14. So how can we simplify them? Remember, fractions are just division problems. We can rewrite the complex fraction as a division problem with two fractions. This division problem then changes to multiplication by the reciprocal.

15. Simplifying Complex Fractions Rule Any complex fraction Where b ≠ 0, c ≠ 0, and d ≠ 0, may be expressed as:

16. What if we have mixed numbers in the complex fraction? If we have mixed numbers, we treat it as an addition problem with unlike denominators. We want to be working with two fractions, so make sure the numerator is one fraction, and the denominator is one fraction Now we can rewrite the complex fraction as a division of two fractions

17. Example

18. Try on your own…

19. What about complex rational expression? Treat the complex rational expression as a division problem Add any rational expressions to form rational expressions in the numerator and denominator Factor Simplify “Bad” values - Extraneous Roots

20. Ex. 2: Simplify ← The LCD is xy for both the numerator and the denominator. ← Add to simplify the numerator and subtract to simplify the denominator. ← Multiply the numerator by the reciprocal of the denominator.

21. Ex. 2: Simplify ← Eliminate common factors.

22. Example

24. Try on your own

25. One more for you

26. Ex. 3: Simplify ← The LCD of the numerator is x + 4, and the LCD of the denominator is x – 3.

27. Ex. 3: Simplify ← FOIL the top and don’t forget to subtract the 1 and add the 48 on the bottom.

28. Ex. 3: Simplify ← Simplify by subtracting the 1 in the numerator and adding the 48 in the denominator.

29. Ex. 3: Simplify ← Multiply by the reciprocal. x 2 + 8x +15 is a common factor that can be eliminated.

30. Ex. 3: Simplify ← Simplify

31. Factor the sum or difference of two cubes The product of the same three factors is called a ___________________________. KNOW THESE….. a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) Factor x 3 – 8 This is x 3 – 2 3, so: x 3 – 8 = x 3 – 2 3 = (x – 2)(x 2 + 2x + 2 2 ) = (x – 2)(x 2 + 2x + 4) Factor 27x 3 + 1 Remember that 1 can be regarded as having been raised to any power you like, so this is really (3x) 3 + 1 3. 27x 3 + 1 = (3x) 3 + 1 3 = (3x + 1)((3x) 2 – (3x)(1) + 1 2 ) = (3x + 1)(9x 2 – 3x + 1)

32. Ex: Answer the following questions 1.How many separate pieces does this function have? 2.List the three equations 3.For equation, list the x intervals for which it is valid. (use inequality notation) 4.How do we write this as a piecewise function? 5.What is the domain of the function? D = (-∞, 6] R = [0, ∞)

33. Model Problems

34. Rule of 4 The Rule of 4 refers to representing mathematical functions with graphs, tables, equations, and words. As learners discover how to represent functions in each of these ways, the mathematics becomes more meaningful. For example, consider the following cell phone plan offered by T-Mobile in 2011, represented using the Rule of 4. 1.Words Representation (from website) 3. Graph Representation Even More 1000 Talk + Unlimited Text $59.99 includes 1000 whenever minutes Additional minutes $0.45 per minute 2. Table Representation 4. Equation Representation Despite the fact that each of these representations of the cell phone cost function looks different, the same function is represented in each representation. All learners should practice to increase their ability to “see” the other forms mentally even when only one form is given.

35. Absolute Value

36. | x | = 5 Absolute Value Equations x = 5 x = – 5 SameOpposite | x | = –2 No Solution Two Solutions Absolute Value Property If |x| = a, where x is a variable or an expression and a  0, then x = a or x =  a.

37. SolvingAbsolute Value Equations Solving Absolute Value Equations 1. Isolate the absolute value so that the equation is in the form |ax + b| = c.  If c < 0, the equation has no solution. 2. Separate the absolute value into two equations, ax + b = c and ax + b =  c. 3. Solve both equations. 3. Check your answers. Make sure they are not extraneous.

38. Absolute Value Equations with 2 Absolute Values Same Opposite Check your work!

39. Absolute Value Equations Same Opposite Drop the absolute value bars! Keep the absolute value bars! 1. Isolate 2. Two Cases 3. Solve Check your work!

40. Absolute Value Equations with 2 Absolute Values Same Opposite Check your work!

41. Absolute Value Equations Same Opposite Drop the absolute value bars! Keep the absolute value bars! 1. Isolate 2. Two Cases 3. Solve Check your work!

42. Absolute Value Equations 1. Isolate 2. Two Cases 3. Solve No Solution

43. Absolute Value Equations Same Opposite 1. Isolate 2. Two Cases 3. Solve Check your work!

44. Absolute Value Equations with 2 Absolute Values Same Opposite No Solution

45. Factoring

47. Factoring a polynomial means expressing it as a product of other polynomials.

48. Factoring polynomials with a common monomial factor (using GCF). **Always look for a GCF before using any other factoring method. Factoring Method #1

49. Steps: 1. Find the greatest common factor (GCF). 2. Divide the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as the product of the quotient and the GCF.

50. Step 1: Step 2: Divide by GCF

51. The answer should look like this:

52. Factor these on your own looking for a GCF.

53. Factoring polynomials that are a difference of squares. Factoring Method #2

54. A “Difference of Squares” is a binomial ( *2 terms only*) and it factors like this:

55. To factor, express each term as a square of a monomial then apply the rule...

56. Here is another example:

57. Try these on your own:

58. End of Day 1

59. Sum and Difference of Cubes:

60. Rewrite as cubes Write each monomial as a cube and apply either of the rules. Apply the rule for sum of cubes:

61. Rewrite as cubes Apply the rule for difference of cubes:

62. Factoring Method #3 Factoring a trinomial in the form: where a = 1

63. Next Factoring a trinomial: 2. Find the factors of the c term that add to the b term. For instance, let c = d·e and d+e = b then the factors are (x+d)(x +e ).. 1. Write two sets of parenthesis, (x )(x ). These will be the factors of the trinomial.

64. x x Factors of +8: 1 & 8 2 & 4 -1 & -8 -2 & -4 Factors of +8 that add to -6 2 + 4 = 6 1 + 8 = 9 -2 - 4 = -6 -1 - 8 = -9 -2 -4

65. Check your answer by using FOIL FOIL

66. Lets do another example: Find a GCF Factor trinomial Don’t Forget Method #1. Always check for GCF before you do anything else.

67. When a>1, let’s do something different! Step 1: Multiply a · c Step 2: Find the factors of a·c (-30) that add to the b term = - 30

68. Factors of 6 · (-5) : 1, -30 1+-30 = -29 -1, 30 -1+30 = 29 2, -15 2+-15 =-13 -2, 15 -2+15 =13 3, -10 3+ -10 =-7 -3, 10 -3+ 10 =7 5, -6 5+ -6 = -2 -5, 6 -5+6 =1 Step 2: Find the factors of a·c that add to the b term Let a·c = d and d = e·f then e+f = b d = -30 e = -2 f = 15

69. -2, 15 -2+15 =13 Step 3: Rewrite the expression separating the b term using the factors e and f Step 4: Group the first two and last two terms.

70. Step 4: Group the first Two and last two terms. Step 5: Factor GCF from each group Check!!!! If you cannot find two common factors, Then this method does not work. Step 6: Factor out GCF Common factors

71. Step 3: Place the factors inside the parenthesis until O + I = bx. FOIL O + I = 30 x - x = 29x This doesn’t work!! Try: I am not a fan of guess and check!

72. FOIL O + I = -6x + 5x = -x This doesn’t work!! Switch the order of the second terms and try again.

73. Try another combination: FOIL O+I = 15x - 2x = 13x IT WORKS!! Switch to 3x and 2x

74. Factoring Technique #3 continued Factoring a perfect square trinomial in the form:

75. Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula!

76. Does the middle term fit the pattern, 2ab? Yes, the factors are (a + b) 2 : b a

77. Does the middle term fit the pattern, 2ab? Yes, the factors are (a - b) 2 : b a

78. Factoring Technique #4 Factoring By Grouping for polynomials with 4 or more terms

79. Factoring By Grouping 1. Group the first set of terms and last set of terms with parentheses. 2. Factor out the GCF from each group so that both sets of parentheses contain the same factors. 3. Factor out the GCF again (the GCF is the factor from step 2).

80. Step 1: Group Example 1: Step 2: Factor out GCF from each group Step 3: Factor out GCF again

81. Example 2:

82. Try these on your own:

83. Answers:

84. t (months)1234567 S(t) (100’s)1.541.882.323.123.784.906.12 The number of new software products sold is given by S(t), where S is measured in hundreds of units and t is measured in months from the initial release date of January 1. 2012. The software company recorded these sales data: a)Estimate the number of units sold between April 1, 2012 through June 30, 2012. b)Assuming the data is linear, determine a the prediction equation estimating the number of units sold during months April and June. c)Using your prediction equation, predict the number of units sold in March and July? d)Graph the data and draw a quick sketch. e)Is your prediction equation accurate, why might it not be accurate? (2x+5)= 4x 2 +16x+15

85. t (months1234567 St (100’s)1.541.882.323.123.784.906.12 The number of new software products sold is given by S(t), where S is measured in hundreds of units and t is measured in months from the initial release date of January 1. 2012. The software company recorded these sales data: a)Estimate the number of units sold between April 1, 2012 through June 30, 2012. Using the table data (4, 5, 6), there were 1180 units sold b) Assuming the data is linear, determine a the prediction equation estimating the number of product sold during months April and June. Using the point (4, 3.12) and (6, 4.9), the equation of the line is: y - 3.12 =.9(x-4) +2 pts c)Using your prediction equation, predict the number of units sold in March and July? Evaluating the equation found in step (b), the number of units sold are for March x = 3 y = 2.22 units and for July x = 7, y = 4.92 units d) Graph the data and draw a quick sketch of your prediction equation. The red line is the graph of the original table data. The green line is the graph of the prediction equation found in (b). The graph was generated using Excel. e)Explain whether your prediction equation is accurate? What might make it be more accurate? Our initial assumption that the data was linear was incorrect. We can look at other regressions that might match the data more accurately. Challenge: Using your calculator, which regression yields the best fit? a)1 pt b) 2 pts c)1 pts d) 2 pts 6) 5 pts

86. Rule of 4 The Rule of 4 refers to representing mathematical functions with graphs, tables, equations, and words. As learners discover how to represent functions in each of these ways, the mathematics becomes more meaningful. For example, consider the following cell phone plan offered by T-Mobile in 2011, represented using the Rule of 4. 1.Words Representation (from website) 3. Graph Representation Even More 1000 Talk + Unlimited Text $59.99 includes 1000 whenever minutes Additional minutes $0.45 per minute 2. Table Representation 4. Equation Representation Despite the fact that each of these representations of the cell phone cost function looks different, the same function is represented in each representation. All learners should practice to increase their ability to “see” the other forms mentally even when only one form is given.

87. Studying Process What does it mean to study in Mrs. Harton’s Class? Studying begins At Home Before coming to class: Read the section and start your notes. Write down the definitions of words and copy theorems. Work through each example. Make note of what you don’t understand. Studying continues In Class Listen to Teacher led instruction Take Notes Ask Questions Do in class work Studying continues with Homework To do Assigned Homework Write Header -Name, Date, Section number, page numbers, problem numbers Copy Question and figures Look at notes and the text to see how to apply definitions and theorems Mark up the figure and solve the problem. Studying continues In Class: Request to see problems you don’t understand and make corrections to Homework

88. Quizzes and Test Show your work. This means I would like to see the set up, the method used and the solution. Sometimes, no work is needed. However, explain your reasoning, why are you able to do what you are doing? If a formula is used, mention it.