# Quadratic Equations in 3 Forms Standard Form Factored

• Slides: 7

Quadratic Equations in 3 Forms: Standard Form, Factored Form, and Vertex Form

y = 4 x 2 + 16 x + 6 y = -x 2 – 8 x + 1

Factored Form of the quadratic equation: f(x) = a(x – r 1) (x – r 2) • where (r 1, 0) and (r 2, 0) are the roots / x-intercepts • a determines if parabola opens up or down and how wide or narrow it will be • It is easy to spot the x-intercepts (roots/zeros) from this form (but not the y intercept or vertex) y = -2(x – 4)(x + 3) y = 1/2(x + 2)(x-9) Roots: (4, 0) & (-3, 0) opens down / has a maximum Roots: (-2, 0) & (9, 0) opens up / has a minimum

Factored Form y=a(x – r 1)(x – r 2) Without graphing, state the: where (r 1, 0) & (r 2, 0) are the x-ints 1. zeros / roots 2. if it will have a maximum or minimum 1. y = 3(x + 6)(x – 2) 1. Zeros / Roots: (-6, 0) (2, 0) 2. Has a minimum 3. y = -9(2 x + 5)(3 x – 1) 1. Zeros / Roots: -5/2, 0) (1/3, 0) 2. Has a maximum 2. y = -2(x – 4)(x – 14) 1. Zeros / Roots: (4, 0) (14, 0) 2. Has a maximum 4. y = (x – 11)(4 x + 9) 1. Zeros / Roots: (11, 0) (-9/4, 0) 2. Has a minimum (

Vertex Form of the quadratic equation: 2 f(x) = a(x – h) + k • where (h, k) are the coordinates of the vertex • a determines if parabola opens up or down and how wide or narrow it will be • It is easy to spot the vertex from this form (but not the intercept or x-intercepts) y = (x – 2)2 + 6 y = -3(x + 2)2 -5 y- Vertex (2, 6) opens up / 6 is minimum value Vertex (-2, -5) opens down / -5 is maximum value

Vertex Form vertex form y = a(x – h)2 + k where (h, k) is the vertex. Without graphing, state the: 1. vertex 2. if it will have a maximum or minimum 3. the minimum/maximum value of the function 1. y = (x – 5)2 + 3 1. Vertex (5, 3) 2. Has a minimum 3. Minimum y value of function: 3 2. y = -x 2 – 2. 5 1. Vertex (0, -2. 5) 2. Has a maximum 3. Maximum y value of function: -2. 5 3. y = -2(x + 4)2 1. Vertex (-4, 0) 2. Has a maximum 3. Maximum y value of function: 0 4. y = (x + 9)2 – 6 1. Vertex (-9, -6) 2. Has a minimum 3. Minimum y value of function: -6

Write a quadratic, in vertex form, whose graph will have the given characteristics: Write a quadratic, in factored form AND standard form, whose graph will have the given zeros: 1. (1. 9, -4) and opens down 1. (1. 9, 0) (4, 0) 1. y = -(x – 1. 9)2 – 4 2. (0, 100) and opens up 1. y = x 2 + 100 (notice this is also the y-int) 3. (-2, 3/ 2) opens down 1. y = -3(x + 2)2 + 3/2 1. y = (x – 1. 9)(x – 4) FOIL factored 2. y = x 2 – 5. 9 x + 7. 6 form to turn into standard form 2. (100, 0) (-10, 0) 1. y = (x – 100)(x + 10) 2. y = x 2 – 90 x - 1000 3. (-2, 0) (3/2, 0) 1. y = (x + 2)(2 x – 3 ) 2. y = 2 x 2 + x – 6