1. Getting Ready: Zero Product Property If two numbers multiply together to equal zero, one or both of the numbers must equal zero. ie) m x n = 0  m or n must be equal to 0 3.3 Factored Form of a Quadratic Relation

2. Determine the zeros (x-intercepts) of the following: RECALL: For x-intercepts, y = 0 a)y = x(x - 20) x – 20 = 0 x = 20x = 0 The zeros are at 0 and 20 0 = x(x - 20)

3. Determine the zeros of the following: b) y = 2(x + 3)(x – 8) 0 = 2(x + 3)(x – 8) x – 8 = 0 x = 8 x + 3= 0 The zeros are at -3 and 8 x = -3 What do you notice?

4. Quadratic equations can be modeled in the form: y = a(x – s)(x – t) where a  0. This is called FACTORED FORM. If a > 0 then the parabola opens up. If a < 0 then the parabola opens down. minimum maximum

5. y = a(x – s)(x – t) Axis of symmetry: Vertex: Zeros (x-intercepts) are s and t

6. Example 1: Given: y = x(x – 4) a) Determine the zeros of the parabola. Let y = 0 0 = x(x – 4) The zeros are x = 0 and x = 4.

7. Given: y = x(x – 4) b) Determine the equation of the axis of symmetry. The axis of symmetry is midway between the zeros. c) Determine the vertex of the parabola. Substitute x = 2into the original equation. x = 0 + 4 2

8. Given: y = x(x – 4) y = (2)(2 – 4) y = (2)(– 2) y = – 4 The vertex is (2, – 4)

9. Sketch the graph of: y = – (4 – x)(8 – x) Example 2: The zeros are at 4 and 8. The axis of symmetry is at x = 6 Substitute x = 6 in the equation to determine the y value of the vertex. y = – (4 – 6)(8 – 6) y = –(– 2)(2) y = 4 The vertex is at (6, 4).

10. If we know the zeros and the a-value, we can write the equation of a quadratic in factored form. Eg 3) Write the equation of the following parabola. Zeros: -6 and 2 Vertex: (-2, 4) y = a(x +6)(x – 2) 4 = a(-2 + 6)(-2 – 2) 4 = a(4)(-4) 4 = -16a – 1 = a 4 y = -1/4(x +6)(x – 2)

11. Determine the equation of the parabola whose x- intercepts are 2 and – 3 and whose y-intercept is 6. Example 4 y = a(x – s)(x – t) y = a(x – 2)(x + 3) sub s = 2 and t = – 3 To find a, substitute (0,6) into the equation. 6 = a(0 – 2)(0 + 3) 6 = a(– 2)(3) 6 = a(– 6) – 1 = a y = – (x – 2)(x + 3)