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Algebra-2 Lesson 4-3A (Intercept Form) Quiz 4-1, 4-2 1. What is the vertex of: f ( x) 2 x 4 x 6 2 2. What is the vertex of: f ( x) ( x 7) 6 2 4-3A Intercept Form y ax bx c 2 Standard Form: y 2 x 2 12 x 1 Axis of symmetry: Vertex: (1) b (12) x x x 3 2a 2(2) b x 2a x 3 y 2(3) 2 12(3) 1 y 17 (2) “2nd” “calculate” “min/max” x-intercepts: “2nd” “calculate” “zero” Vertex Form: Vertex: y a ( x h) k 2 (h, k) (1) y 2( x 1)2 3 (1, 3) (2) “2nd” “calculate” “min/max” Axis of symmetry: x-intercepts: xh x 1 “2nd” “calculate” “zero” Vocabulary Intercept Form: y a( x p)( x q) y 4( x 1)( x 3) Intercept form Graph the following on your calculator: y ( x 1)( x 2) x = -1 x = +2 What are the x-intercepts? y=0 Vocabulary Opens up if positive Intercept Form: y 3( x 2)( x 4) Opens down ‘x-intercepts are: ‘-2’ and ‘-4’ ‘x-intercepts are: ‘p’ and ‘q’ y a( x p)( x q) y ( x 1)( x 3) ‘x-intercepts are: ‘+1’ and ‘+3’ Intercept form Why do the intercept have the opposite sign? y ( x 1)( x 2) x = -1 x = +2 0 ( x 1)( x 2) (x + 1) equals some number. (x – 2) equals another number. These two numbers multiplied together equal 0. (x + 1) = 0 x = -1 (x – 2) = 0 x = +2 y=0 Vocabulary Zero Product Property: If the product of 2 numbers equals 0, A*B=0 then either: A=0 and/or 0 ( x 2)( x 3) Then by the zero product property: x20 x 2 x 3 0 x 3 B = 0. Your turn: y a( x p)( x q) Which direction does it open and what are the x-intercepts of the the following parabolas: 1. y 5( x 3)( x 4) 2. y 2( x 7)( x 6) 3. y ( x 2)( x 5) y a( x p)( x q) Finding the vertex: If you know the x-intercepts, how do you find the axis of symmetry? Half way between the x-intercepts. y 2( x 4)( x 6) 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? y 2(( ) 3)(( ) 4) y 2(2)(1) The vertex is: y 2((5) 3)((5) 4) y4 (5, 4) Your turn: y a( x p)( x q) Find the vertex of the parabola: 4. y ( x 2)( x 6) 5. y 2( x 2)( x 4) 6. y ( x 3)( x 5) Vocabulary Monomial: an expression with one term. 2x Binomial: expression with two unlike terms. x 1 The sum (or difference) of 2 unlike monomials. Vocabulary Trinomial: expression with three unlike terms. The sum of 3 unlike monomials x 3x 2 2 Or the product of 2 binomials. ( x 2)( x 1) Intercept form is the product of 2 binomials!! y ( x p)( x q) y ( x 2)( x 1) Product of Two Binomials Know how to multiply two binomials (x – 5)(x + 1) Distributive Property (two times) x(x + 1) – 5(x + 1) x x 5x 5 2 x2 4x 5 Product of Two Binomials Know how to multiply two binomials (x – 3)(x + 2) Distributive Property (two times) x(x + 2) – 3(x + 2) x 2 x 3x 6 2 x2 x 6 Your turn: Multiply the following binomials: 7. 8. 9. ( x 2)( x 6) ( x 2)( x 4) ( x 3)( x 5) Taught to here as 4-3A Your turn: Multiply the following binomials: 1. ( x 2)( x 6) 2. ( x 2)( x 4) 3. ( x 3)( x 5) I call this method the “smiley face”. You have learned it as FOIL. Smiley Face x 2 4x 2x 8 x 2x 8 2 (x – 4)(x + 2) = ? Left-most term left “eyebrow” “nose and chin” combine to form the middle term. right-most term right “eyebrow” Your turn: Multiply the following binomials: 4. ( x 1)( x 7) 5. ( x 3)( x 2) 6. ( x 3)( x 3) Convert Intercept Form to Standard Form y a( x p)( x q) y ax bx c 2 Just multiply the binomials. y ( x 1)( x 7) y x( x 7) 1( x 7) y x 7x x 7 2 y x 8x 7 2 Vocabulary To Factor: split a binomial, trinomial (or any “nomial”) into its original factors. Standard form: Factored form: y ax bx c y a( x p)( x q) y x 2x 1 y ( x 2)( x 1) 2 2 Intercept form is a standard form that has been factored. Factoring Quadratic expressions: x 5x x 5 2 (x – 5)(x + 1) x 4x 5 2 (_ + _)(_ + _) x 4x 5 2 Factoring Quadratic expressions: x 5x x 5 2 x2 4x 5 (x – 5)(x + 1) = ? x2 4x 5 (x + _)(x + _) -1, 5 1, -5 -1, 5 5, -1 -5, 1 1, -5 Factoring Quadratic expressions: 2 x 4 x 5 (x – 5)(x + 1) = ? x 4x 5 2 (x + _)(x + _) (x – 5)(x + 1) -1, 5 1, -5 (x – 1)(x + 5) (x – 5)(x + 1) Factoring x bx c (x m)(x n) 2 c = mn b=n+m x (m n) x mn 2 x 5x 6 2 (x + 3)(x + 2) What 2 numbers when multiplied equal 6 and when added equal 5? Factoring x bx c x (m n) x mn (x m)(x n) 2 2 2 x 4x 5 (x – 5)(x + 1) What 2 numbers when multiplied equal -5 and when added equal -4? Factoring x 6x 8 2 What 2 numbers when multiplied equal 8 and when added equal -6? (x – 2)(x – 4) Your Turn: Factor: 7. x 4x 3 8. x 2x 1 9. 2 2 x 6x 9 2 They come in 4 types: Both positive 1st Negative, 2nd Positive x 4x 3 x 6x 5 (x + 3)(x + 1) (x – 1)(x – 5) 2 Both negative 2 1st Positive, 2nd Negative x 2x 8 x 6 x 16 (x – 4)(x + 2) (x + 8)(x – 2) 2 2 Your Turn: 10. 11. Factor: x 6x 5 2 x 6 x 16 2 12. x 2x 8 13. x 4 x 12 2 2 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. 0 AB If A= 5, what must B equal? If B = -2, what must A equal? Zero Product Property 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. f ( x) ax bx c 2 Solve by factoring y x 3x 2 2 (1) factor the quadratic equation. (2) set y = 0 y ( x 2)( x 1) 0 ( x 2)( x 1) (3) Use “zero product property” to find the x-intercepts ( x 2) 0 and x 2 ( x 1) 0 x 1 f ( x) ax bx c 2 Solve by factoring y x 5x 6 2 (1) factor the quadratic equation. (2) set y = 0 y ( x 2)( x 3) 0 ( x 2)( x 3) (3) Use “zero product property” to find the x-intercepts ( x 2) 0 and x2 ( x 3) 0 x 3 Your Turn: Solve by factoring: 2 14. y x 9 x 14 15. y x 7x 8 16. y x 8 x 16 2 2 What if it’s not in standard form? 2 x 17 7 11x Re-arrange into standard form. x 11x 24 0 2 ( x 3)( x 8) 0 3 + 8 = 11 x = -3 3 * 8 = 24 x = -8 Your Turn: 17. 18. Solve by factoring: 2x x 3 x 6x 9 2 2 3x 2 x 10 2 x 8 x 6 2 2 What if the coefficient of ‘x’ ≠ 1? Solve by factoring: 0 (2 x 4)(9 x 3) Use “zero product property” to find the x-intercepts 2x 4 0 and 9x 3 0 9x 3 2x 4 3 x x2 9 1 x 3 Your Turn: Solve 19. 20. y (2 x 4)( x 14) 0 ( x 7)(3x 2) Your turn: Multiply the binomials: 21. (2x – 1)(x + 3) 22. (x + 5)(x – 5) Factor the quadratic expressions: 2 2 26. x 100 25. 4 x 64 26. x 2x 8 2 27. x 4 4 Special Products Product of a sum and a difference. (x + 2)(x – 2) “conjugate pairs” (x + 2)(x – 2) x 2x 2x 4 2 “nose and chin” are additive inverses of each other. “The difference of 2 squares.” x 4 2 ( x) (2) 2 2 Your turn: Multiply the following conjugate pairs: 13. 14. (x – 3)(x + 3) (x – 4)(x + 4) “The difference of 2 squares.” x 9 2 x 16 “The difference of 2 squares” factors as conjugate pairs. 2 Your Turn: Factor: 15. x 2 36 16. x 49 2 Special Products ( x 2) Square of a sum. 2 2 x 2x 2x 2 2 (x + 2)(x + 2) x 4x 4 2 Special Products ( x 3) Square of a sum. x 3x 3x 3 2 (x + 3)(x + 3) 2 2 x 6x 9 2 Special Products Square of a difference. x 4x 4x 4 2 ( x 4) 2 2 (x - 4)(x - 4) x 8 x 16 2 Special Products Square of a difference. x 3x 3x 4 2 ( x 3) 2 2 (x - 3)(x - 3) x 6 x 9 2 Your Turn: Simplify (multiply out) 17. ( x 4) 18. ( x 6) 2 2 We now have all the tools to “solve by factoring” Vocabulary Quadratic Equation: f ( x) ax bx c 2 f ( x) x x 6 2 Root of an equation: the x-value where the graph 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.