3.3 Factored Form of a Quadratic Relation
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
Determine the zeros (x-intercepts) of the following:
RECALL: For x-intercepts, y = 0
a) y = x(x - 20)
0 = x(x - 20)
x=0
x – 20 = 0
x = 20
The zeros are at 0 and 20
Determine the zeros of the following:
b) y = 2(x + 3)(x – 8)
0 = 2(x + 3)(x – 8)
x + 3= 0
x = -3
x–8=0
x=8
The zeros are at -3 and 8
What do you notice?
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
If a > 0 then the
parabola opens down.
parabola opens up.
maximum
minimum
y = a(x – s)(x – t)
Zeros (x-intercepts) are s and t
Axis of symmetry:
Vertex:
 st 
, y

 2

st
x
2
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.
Given: y = x(x – 4)
b) Determine the equation of
the axis of symmetry.
The axis of symmetry is
midway between the zeros.
x= 0+4
2
c) Determine the vertex of
the parabola.
Substitute x = 2 into the
original equation.
Given: y = x(x – 4)
y = (2)(2 – 4)
y = (2)(– 2)
y=–4
The vertex is (2, – 4)
Example 2:
Sketch the graph of: y = – (4 – x)(8 – x)
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).
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)
Example 4
Determine the equation of the parabola whose xintercepts are 2 and – 3 and whose y-intercept is 6.
y = a(x – s)(x – t)
sub s = 2 and t = – 3
y = a(x – 2)(x + 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)