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

2. 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.

3. 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

4. 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

5. 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

6. 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

7. 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
(-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 = x
(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