Quadratic Functions and Parabolas
General Form:
Note:
Standard Form:
, where
and
are the coordinates of the vertex.
Quadratic Formula:
The radicand
is called the discriminant
.
The Discriminant Test determines the number and type of roots (solutions) in the parabola.
If
, there is exactly one repeated real root.
If
, there are two (2) distinct real roots.
If
, there are two (2) complex/imaginary roots (example:
).
Example:
= 4,
=
8,
=3
So, there are two distinct real roots.
To find the two distinct real roots, use the quadratic formula to solve for :
.
Note that in this case, the equation can also be solved by factoring:
=½
or
= 3/2
Graphically, (1/2, 0) and (3/2, 0) are the -intercepts of the parabola
.
The value of has the following attributes:
If > 0, parabola opens UP (smiles)
If < 0, parabola opens DOWN (frowns)
If
, parabola opens NARROWER than
If
and ≠ 0, parabola opens WIDER than
Vertex
: The vertex is the turning point of a parabola.
is the -coordinate of the vertex and is the -coordinate of the vertex.
The vertex is a minimum if the parabola opens up and a maximum if it opens down.
In general form
In standard form
,
and
and
.
,
are shown in the equation.
Example:
Example:
and
.
So, the vertex is at
.
So, the vertex is at (3, 5).
Note: has the opposite sign of the number
inside the perfect square.
Lone Star College – Montgomery Learning Center: Quadratics and Parabolas
Updated April 7, 2011
Page 2 of 3
Axis of Symmetry: A vertical line passing through the vertex with the equation
Finding and intercepts:
-intercept(s):
In General Form, solve
by factoring or the quadratic formula.
In Standard Form, solve
by taking square roots on both sides.
-intercept:
Substitute 0 for and find .
Example: Find vertex, axis of symmetry and intercepts of
Solution:
. Then graph the function.
Vertex:
Axis of Symmetry:
To find -intercepts, solve
.
.
-intercepts are at
and
To find -intercept,
-intercept is at
.
.
.
, parabola opens up
Lone Star College – Montgomery Learning Center
Updated April 7, 2011
.
Page 3 of 3