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Algebra II

Algebra II. Unit 2. Years Since 1985. 0. 3. 6. 9. 12. 15. Percent. 20. 18. 15. 15. 14. 13. Source: U.S. News & World Report. Preview . 1. Write the equation of the line passing through (-2, 3) and (4, 0) 2. Create a scatter plot for the data at right. 3. Draw a trend line

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Algebra II

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  1. Algebra II Unit 2

  2. Years Since 1985 0 3 6 9 12 15 Percent 20 18 15 15 14 13 Source: U.S. News & World Report Preview • 1. Write the equation of the line passing through (-2, 3) and (4, 0) • 2. Create a scatter plot for the data at right. • 3. Draw a trend line (line of best fit). • 4. For the scatter plot, what kind of relationship do you see? (positive, negative or no corelation?) Percent of students sending applications to two colleges

  3. Preview • 1. 0 – 3 = - ½ 4 – (-2) 0 = - ½ (4) + b y = - ½ x + 2 OR y – 3 = - ½ ( x - - 2) y – 3 = - ½ x – 1 y = - ½ x + 2 • 4. negative 1) Find slope 2) Find b and/or equation

  4. Unit 2.1 (Glencoe Chapter 2.5) Algebra II Modeling Real World Data

  5. I. Write a Prediction Equation • A. Prediction equation: 1. also known as a regression equation 2. use two points and write a slope- intercept equation • Use points what best fit the data (are solid with the line of best fit)

  6. Years Since 1985 0 3 6 9 12 15 Percent 20 18 15 15 14 13 Source: U.S. News & World Report I. Write a Prediction Equation Percent of students sending applications to two colleges • B. Example: • 1. Use points (3, 18) and (15, 13) because they are solid points • 2. Find slope: 13 – 18 = -5 or - 0.42 15 – 3 12 • 3. Find y-intercept point (“b) use one point: 18 = -0.42(3) + b so 19.26 = “b” • 4. Write an equation in y = mx + b form: y = -0.42x + 19.26

  7. I. Write a Prediction Equation • C. Example: y = -0.42x + 19.26 • 1. What is the real meaning of the – 0.42? • 2. Does the 19.26 fit with the line of best fit on our graph? • Your equation will always be in y = mx + b form

  8. II. Make a Prediction Equation • A. Example: y = -0.42x + 19.26 • Use the regression equation (Prediction Equation) to make a prediction: • 1. What will the percent be in year 18? y = -0.42(18) + 19.26 = 11.7 • 2. What will the percent be in 2010? (HINT: what number year will that be?)

  9. Let’s Review the process: The Prediction Equation Then use two good points to write an equation in y = mx + b form. Then predict by substituting an “x” value. This is real data: You may round!

  10. Practice Today: • You may work in your working groups: • Unit 2.1 (Practice 2.5) Handout • #1 (use (1.4, 24) and (2.4, 15) • and # 2 use (7500, 61) and (9700, 50) 1. Create a scatter plot 2.Write a prediction equation 3.Predict the missing values

  11. Check It Out! • 1. b) Prediction equation: y = - 9x + 36.6 • c) Prediction: 18.6 miles per gallon • 2. b) Prediction equation: y = - 0.005x + 98.5 • c) Prediction: 38.5º F

  12. III. Use Technology • Handout • Follow instructions in your groups • Complete all parts of the handout. • Every student turns in a handout!

  13. Systems of Equations Unit 2.2 (Glencoe Chapter 3.1-3.3) Algebra II

  14. I. Review Systems of Two Equations • Solve. Use any method. • 1. x + 2y = 10 x + y = 6 • 2. 2x + 3y = 12 5x – 2y = 11 (2, 4) (3, 2)

  15. I. You Try: Take Five • Solve using any method: 1. 2p + 4q = 18 3p – 6q = 3 2. 3x – y = 4 y = 4 + x 3. y = 3x y = - 2x + 5

  16. II. Solve Systems of Two Equations by Graphing • 1. y = -2x + 5 y = x – 1 • 2. x – 2y = 0 x + y = 6

  17. Check it out: Calculator Tips • In y= menu, enter equation 1 as y1 and equation 2 as y2 • Graph (remember zoom ► 6) • Find intersection: 2nd trace (calc) ►5 (intersect) enter enter enter enter • Write as ordered pair ( x, y)

  18. II. You Try: Take Five • 1. 2x + 3y = 12 2x – y = 4 • 2. 2x – y = 6 x – 8y = - 12

  19. II. How did you do? • 1. 2x + 3y = 12 2x – y = 4 • 2. 2x – y = 6 x – 8y = - 12

  20. For Your Practice • Page 122 • Practice Quiz 1 • #1 – 4

  21. Bellwork 2-04-10 • Page 120 #14, 22 • Word problem: Practice Quiz 1 Page 122 #5! Page 120 #14 (2, 7) #22 (3, -1) Page 122 #5 Hartsfield 78 million O Hare 72.5 million

  22. Systems of Equations: Part 2 Unit 2.2 Glencoe Chapter 3.5 Algebra II

  23. II. Systems of Three Equations (Three Variables) Make a plan before doing anything! • A. One solution (x, y, z) • 1. 5x + 3y + 2z = 2 2x + y – z = 5 x + 4y + 2z = 16 • Use elimination or substitution to create a system of 2 variables: 5x + 3y + 2z = 2 2(2x + y – z = 5) 2(2x + y – z = 5 )x + 4y + 2z = 16 5x + 3y + 2z = 2 4x + 2y – 2z = 10 4x + 2y – 2z = 10x + 4y + 2z = 16 9x + 5y = 12 5x + 6y = 26 Use 2nd and 3rd and Eliminate z! Use 1st and 2nd and Eliminate z!

  24. III. Systems of Three Equations (Three Variables) • A. One solution (x, y, z) • 1. 9x + 5y = 12 5x + 6y = 26 • Now solve the new system: 6(9x + 5y = 12) -5(5x + 6y = 26) 54x + 30y = 72 -25x – 30y = -130 29x = - 58 so, x = - 2 Find y: 9(-2) + 5y = 12 -18 + 5y = 12 so, y = 6 Find z: -2 + 4(6) + 2z = 16 22 + 2z = 16 so, z = - 3 (-2, 6, -3)

  25. III. Systems of Three Equations x + 2y = 12 3y – 4z = 25 X + 6y + z = 20 (6, 3, -4)

  26. You Try: Take Ten • Textbook Page 142 • # 5, 7

  27. How did you do? • 5. (-1, -3, 7) • 7. (5, 2, -1)

  28. BELLWORK 2-05-10 • Page 142 • #12, 14 • #10, 11

  29. Check Page 142 • 12. (3, 4, 7) • 14. (2, -3, 6) • 10. 6c + 3s + r = 42 (cost) c + s + r = 13 ½ (pounds) r = 2s (rice/sausage) • 11. (4 ½ , 3, 6) 4 ½ lb. chicken, 3 lb. sausage, 6 lb. rice Rice is Bigger!

  30. IV. Systems of Three Equations Infinitely many solutions See page 140 EX. 2 • Special Cases • 1. 2x + y – 3z = 5 x + 2y – 4z = 7 6x + 3y – 9z = 15 • Eliminate x : 2x + y – 3z = 5 -6(x + 2y – 4z = 7) -2(x + 2y – 4z = 7)6x + 3y – 9z = 15 2x + y – 3z = 5 - 6x – 12y + 24z = -42 -2x – 4y + 8z = - 146x + 3y – 9z = 15 - 3y + 5z = - 9 -9 y + 15z = - 27 Are these the same line???? GCF: 3; Solve for 0 = 0 And prove same line

  31. IV. Systems of Three Equations No solution • Special Cases • 2. 3x – y – 2z = 4 6x + 4y + 8z = 11 9x + 6y + 12z = - 3 • Eliminate y: 4(3x – y – 2z = 4) -3(6x + 4y + 8z = 11) 6x + 4y + 8z = 112(9x + 6y + 12z = -3) 12x – 4y – 8z = 16 -18x – 12y – 24z = - 33 6x + 4y + 8z = 1118x + 12y + 24z = - 6 18x = 27 0 = - 39 not a true statement!

  32. Practice 2.1 – 2.3 • Worksheet for Extra Practice

  33. Dividing Polynomials Unit 2.3 (Chapter 5.3) Algebra II Honors

  34. I. Divide Polynomials • (x4 – 2x3 + x2 – 3x + 2) ÷ (x – 2) Is the same as: • x4 – 2x3 + x2 – 3x + 2 x – 2 and • (x4 – 2x3 + x2 – 3x + 2)(x – 2) – 1

  35. I. A. Divide Polynomials:No Remainder • 1. (x4 – 4x3 + x2 + 7x – 2 ) ÷ (x – 2)

  36. I. A. You Try: • (2x4 + 7x3 – 14x2 – x – 30) (x + 5) ÷

  37. I. B. Divide Polynomials:with placeholder • 2. (x4 – 5x3 – 13x2 + 7) ÷ (x +1) You must use a placeholder when a variable is missing!

  38. I. B. Divide Polynomials:(a > 1) ) with placeholder • 3. (4x4 – 17x2 + 14x – 3)(2x – 3)-1 You must use a placeholder when a variable is missing!

  39. Take Ten: You Try • Page 238 Practice Quiz 1 • #9, 10

  40. II.A. Divide Polynomials (Monomial) • 1. 4x3y2 + 8xy2 – 12x2y3 4xy • 2. 5a2b – 15ab3 + 10a3b4 5ab

  41. For Your Practice • TEXTBOOK PAGE 236 – 237 • #17,18, 22, 24, 32 • Copy problems in form set up for division and divide. • Write all remainders as fractions.

  42. II. B. Using Synthetic Division (divisor in form of x – r! • 1. (5x3 – 13x2 + 10x – 8) ÷ (x – 2) 5 - 13 10 - 8 2 5 (you do not subtract, you add because you changed the sign of the divisor…) 10 -6 8 -3 4 0 5x2 – 3x + 4

  43. You Try: • Use Synthetic Division: (x3 + 5x2 – x – 6) ÷ (x + 2)

  44. II. B. Using Synthetic Division (divisor in form of x – r! • 2. (4x4 – 5x2 + 2x + 4) ÷ (2x – 1) 2 0 - 5/2 1 2 ½ Divide everything by 2! 1  2 1 ½ -1 0 Simplify! -2 0 2 2 1 2x3 + x2 – 2x + 2 x - ½ 2x3 + x2 – 2x + 4 2x - 1 Change sign back!

  45. You Try: • Use Synthetic Division: (2x4 + 3x3 – 2x2 – 3x – 6) ÷(2x + 3)

  46. Harder Factoring Unit 2.4 (Chapter 5.4) Algebra II Honors

  47. I. Review:Factor ax2 + bx + c, a ≠ 1 • Factor: Use Guess and Check or Product/Sum • 1. 2a2 + 3a + 1 • 2. 6x2 + 13x + 6

  48. II. Factor by Grouping (4 terms) • Look for GCF in pairs of terms • 1. 8xy + 20x + 6y + 15 • 2. 18x2 – 30x – 3x + 5

  49. II. Factor by Grouping (4 terms) • Look for GCF in pairs of terms: You try. • 3. 8ax – 6x – 12a + 9 • 4. 10x2 – 14xy – 15x + 21y

  50. III. Sum and Difference of Two cubes • Sum of 2 Cubes • a3 + b3 = (a + b)(a2 – ab + b2) • Difference of 2 cubes • a3 – b3 = (a – b)(a2 + ab + b2) • Factor: • 8a3 + 27 = • (2a + 3)(4a2 – 6a + 9)

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