Quadratic Functions
Factoring & Graphing
Vocabulary:
Quadratic, Degree, Coefficient, Trinomial, Binomial,
Constant, Zeroes, Vertex, Axis or Line of
Symmetry, Vertical Intercept (y-intercept)
2
f(x) = x + 7x - 18 = (
2
g(x) = x - 49 = (
2
r(x) = x - 6x + 9 = (
2
h(x) = 3x - 2x - 8 = (
)(
)(
)
)
)(
)(
)
)
Graphing a Quadratic Function
2
f(x) = x + 7x - 18 = (x + 9)(x - 2)
Zeroes: x = -9 and 2
Axis of Symmetry: (-9 + 2)/2 = -7/2
Vertex: f(-7/2)
2
= (-7/2) + 7(-7/2) - 18
= (49/4) - (49/2) - 18
= (49/4) - (98/4) - (72/4)
= -121/4 or -30 1/4
and Vertical Intercept (0, -18)
Graph not to scale
Zeroes (x-intercepts)
9
-7/2
2
Vertical Intercept
(0, -18)
(y-intercept)
(-3 1/2, -30 1/4 )
Vertex
Try these!
2
f(x) = x - 7x + 12
2
f(x) = x -4x - 32
2
f(x) = 3x + 13x - 10
Concavity with Parabolas
It's important to understand the parabolas are
concave up or down. The slopes between different
intervals change and become more or less steep as
you move along the horizontal axis.
Quadratic Formula
Sometimes you won't be able to factor a quadratic
to determine it's graph. (or, it may not be easy!)
You can always use the quadratic formula to find
the zeroes, and even more information about the
parabola.
x=
-b +-
from the standard or general form:
2
b - 4ac
2
f(x) = ax + bx = c
2a
2
So, for f(x) = 3x + 13x - 10
x=
-13 +-
2(3)
a = 3, b = 13 and c = -10
=
-13 +-
169 + 120
6
=
-13 +-
289
6
=
2
(13) - 4(-30)
-13 +- 17
6
=
2 and -5
3
Try these!
2
f(x) = 7x - 17x - 12
2
f(x) - 15x + x -2
x=
-b +-
2
b - 4ac
2a
Factored Form of a Parabola
y = a (x - r)(x - s)
y = a (x - 1)(x - 3) and we know that (0,3) is on the parabola
so, 3 = a (0 - 1)(0-3) will allow us to solve for a.
3 = a (3), so a = 1. The factored form of the equation will
be y = 1 (x - 1)(x - 3), or just y = (x - 1)(x - 3)
Vertex Form of a Parabola
2
f(x) = a (x - h) + k
(h, k) is the vertex
x = h is the axis of symmetry
To convert a quadratic equation to Vertex Form you
need to Complete the Square!
2
Let's start with f(x) = x + 8x - 11
2
f(x) = x + 8x + 16 - 11 - 16
2
= (x + 4) - 27
So, the vertex is (-4, -27)
and the axis of symmetry is x = -4
Try these! Complete the square to change
to vertex form.
2
f(x) = x + 2x - 15
2
f(x) = x - 3x - 28