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

Standard Form of the quadratic equation: f(x) = ax2 + bx + c where (0, c) are the coordinates of the y-intercept a determines if parabola opens up or down, has minimum or maximum value, and how wide or narrow it will be Easy to spot the y-intercept from this form (but not the vertex or x-intercepts) Have to use x= −b 2a to find x value of vertex and plug it in to find y value y = 4x2 + 16x + 6 y-int (0,6) opens up / has a minimum value x= −b 2a = −16 8 = -2, vertex (-2,-10) is min. y = -x2 – 8x + 1 y-int (0,1) opens down / has a maximum value x= −b 2a = 8 −2 = -4, vertex (-4, 17) is max.

f(x) = a(x – r1) (x – r2) Factored Form of the quadratic equation: where (r1, 0) and (r2, 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) Roots: (4,0) & (-3,0) opens down / has a maximum y = 1/2(x + 2)(x-9) Roots: (-2,0) & (9,0) opens up / has a minimum

Factored Form Factored Form y=a(x – r1)(x – r2) where (r1, 0) & (r2, 0) are the x-ints Without graphing, state the: 1. zeros / roots 2. if it will have a maximum or minimum y = 3(x + 6)(x – 2) Zeros / Roots: (-6,0) (2,0) Has a minimum y = -2(x – 4)(x – 14) Zeros / Roots: (4,0) (14,0) Has a maximum y = -9(2x + 5)(3x – 1) Zeros / Roots: (-5/2,0) (1/3,0) Has a maximum y = (x – 11)(4x + 9) Zeros / Roots: (11,0) (-9/4,0) Has a minimum

Vertex Form of the quadratic equation: f(x) = a(x – h)2 + 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 y-intercept or x-intercepts) y = (x – 2)2 + 6 Vertex (2,6) opens up / 6 is minimum value y = -3(x + 2)2 - 5 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 y = (x – 5)2 + 3 Vertex (5, 3) Has a minimum Minimum y value of function: 3 y = -x2 – 2.5 Vertex (0, -2.5) Has a maximum Maximum y value of function: -2.5 y = -2(x + 4)2 Vertex (-4, 0) Has a maximum Maximum y value of function: 0 y = (x + 9)2 – 6 Vertex (-9, -6) Has a minimum 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: (1.9, 0) (4, 0) y = (x – 1.9)(x – 4) y = x2 – 5.9x + 7.6 (100, 0) (-10, 0) y = (x – 100)(x + 10) y = x2 – 90x - 1000 (-2, 0) (3/2, 0) y = (x + 2)(2x – 3 ) y = 2x2 + x – 6 (1.9, -4) and opens down y = -(x – 1.9)2 – 4 (0, 100) and opens up y = x2 + 100 (notice this is also the y-int) (-2, 3/2) opens down y = -3(x + 2)2 + 3/2 FOIL factored form to turn into standard form