1. Algebra-2 Lesson 4-3A (Intercept Form)

2. Quiz 4-1, 4-2 1. What is the vertex of: 2. What is the vertex of:

3. Intercept Form Intercept Form 4-3A

4. Standard Form: Axis of symmetry: Vertex: x-intercepts: (1) (2) “2 nd ” “calculate” “min/max” “2 nd ” “calculate” “zero”

5. Axis of symmetry: Vertex: x-intercepts: (1) (2) “2 nd ” “calculate” “min/max” “2 nd ” “calculate” “zero” Vertex Form:

6. Vocabulary Intercept Form:

7. Intercept form y = 0 x = -1 x = -1 Graph the following on your calculator: on your calculator: What are the x-intercepts? x = +2 x = +2

8. Vocabulary Intercept Form: Opens up if positive ‘x-intercepts are: ‘p’ and ‘q’ ‘x-intercepts are: ‘+1’ and ‘+3’ ‘x-intercepts are: ‘-2’ and ‘-4’ Opensdown

9. Intercept form y = 0 x = -1 x = -1 Why do the intercept have the opposite sign? have the opposite sign? (x + 1) equals some number. x = +2 x = +2 (x – 2) equals another number. These two numbers multiplied together equal 0. (x + 1) = 0 (x – 2) = 0 x = -1 x = -1 x = +2 x = +2

10. Vocabulary Zero Product Property: If the product of 2 numbers equals 0, A * B = 0 then either: A = 0 and/or B = 0. Then by the zero product property:

11. Your turn: Which direction does it open and what are the Which direction does it open and what are the x-intercepts of the the following parabolas: x-intercepts of the the following parabolas: 1. 2. 3.

12. Finding the vertex: If you know the x-intercepts, how do you find the axis of symmetry? Half way between the x-intercepts. x-intercepts are: 4, 6 Axis of symmetry is: x = 5 If you know the axis of symmetry, how do you find the x-coordinate of the vertex? Same as the axis of symmetry x = 5 If you know the x-coordinate of the vertex, how do you find the y-coordinate? The vertex is: (5, 4)

13. Your turn: Find the vertex of the parabola: 4. 5. 6.

14. Vocabulary Monomial: an expression with one term. Binomial: expression with two unlike terms. The sum (or difference) of 2 unlike monomials.

15. Vocabulary Trinomial: expression with three unlike terms. The sum of 3 unlike monomials Or the product of 2 binomials. Intercept form is the product of 2 binomials!!

16. Product of Two Binomials Know how to multiply two binomials (x – 5)(x + 1) x(x + 1) – 5(x + 1) Distributive Property (two times)

17. Product of Two Binomials Know how to multiply two binomials (x – 3)(x + 2) x(x + 2) – 3(x + 2) Distributive Property (two times)

18. Your turn: Multiply the following binomials: 7. 8. 9.

19. Taught to here as 4-3A

20. Your turn: Multiply the following binomials: 1. 2. 3.

21. Smiley Face I call this method the “smiley face”. (x – 4)(x + 2) = ? Left-most term  left “eyebrow” right-most term  right “eyebrow” “nose and chin” combine to form the middle term. You have learned it as FOIL.

22. Your turn: Multiply the following binomials: 4. 5. 6.

23. Convert Intercept Form to Standard Form Just multiply the binomials.

24. Vocabulary To Factor: split a binomial, trinomial (or any “nomial”) into its original factors. “nomial”) into its original factors. Standard form: Factored form: Intercept form is a standard form that has been factored.

25. Factoring Quadratic expressions: (x – 5)(x + 1) (_ + _)(_ + _)

26. Factoring Quadratic expressions: (x – 5)(x + 1) = ? (x + _)(x + _) -1, 5 5, -1 -5, 1 1, -5 -1, 5 1, -5

27. Factoring Quadratic expressions: (x – 5)(x + 1) = ? (x + _)(x + _) -1, 5 1, -5 (x – 1)(x + 5) (x – 5)(x + 1)

28. (x m)(x n) c = mn (x + 3)(x + 2) Factoring What 2 numbers when multiplied equal 6 and when added equal 5? b = n + m

29. (x m)(x n) (x – 5)(x + 1) Factoring What 2 numbers when multiplied equal -5 and when added equal -4?

30. (x – 2)(x – 4) Factoring What 2 numbers when multiplied equal 8 and when added equal -6?

31. Your Turn: Factor: 7.8.9.

32. They come in 4 types: (x + 3)(x + 1) Both positive 1 st Negative, 2 nd Positive (x – 1)(x – 5) 1 st Positive, 2 nd Negative (x + 8)(x – 2) Both negative (x – 4)(x + 2)

33. Your Turn: Factor: 10.11. 12.13.

34. Vocabulary Solution (of a quadratic equation): The input values that result in the function equaling zero. If the parabola crosses the x-axis, these are the x-intercepts.

35. Zero Product Property If A= 5, what must B equal? If B = -2, what must A equal? Zero product property: if the product of two factors equals zero, then either: (a)One of the two factors must equal zero, or (b)both of the factors equal zero.

36. Solve by factoring (1) factor the quadratic equation. (1) factor the quadratic equation. (2) set y = 0 (3) Use “zero product property” to find the x-intercepts

37. Solve by factoring (1) factor the quadratic equation. (1) factor the quadratic equation. (2) set y = 0 (3) Use “zero product property” to find the x-intercepts

38. Your Turn: Solve by factoring: 14.15.16.

39. What if it’s not in standard form? Re-arrange into standard form. 3 + 8 = 11 3 * 8 = 24 x = -3 x = -8 x = -8

40. Your Turn: Solve by factoring: 17.18.

41. What if the coefficient of ‘x’ ≠ 1? Solve by factoring: Use “zero product property” to find the x-intercepts

42. Your Turn: Solve 19.20.

43. Your turn: Multiply the binomials: Multiply the binomials: 21. (2x – 1)(x + 3) 22. (x + 5)(x – 5) Factor the quadratic expressions: 25. 26. 26. 27.

44. Special Products Product of a sum and a difference. (x + 2)(x – 2) “conjugate pairs” (x + 2)(x – 2) “nose and chin” are additive inverses are additive inverses of each other. of each other. “The difference of 2 squares.”

45. Your turn: Multiply the following conjugate pairs: 13. (x – 3)(x + 3) 14. (x – 4)(x + 4) “The difference of 2 squares.” “The difference of 2 squares” factors as conjugate pairs.

46. Your Turn: Factor: 15.16.

47. Special Products Square of a sum. (x + 2)(x + 2)

48. Special Products Square of a sum. (x + 3)(x + 3)

49. Special Products Square of a difference. (x - 4)(x - 4)

50. Special Products Square of a difference. (x - 3)(x - 3)

51. Your Turn: Simplify (multiply out) 17.18. We now have all the tools to “solve by factoring”

52. Vocabulary Quadratic Equation: Root of an equation: the x-value where the graph crosses the x-axis (y = 0). crosses the x-axis (y = 0). Zero of a function: same as root Solution of a function: same as both root and zero of the function. x-intercept: same as all 3 above.