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

Lone Star College – Montgomery Learning Center: Quadratics and Parabolas Page 2 of 3 Updated April 7, 2011

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 ,

and .

Example:

. So, the vertex is at .

In standard form , and are shown in the equation. Example:

and

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 Page 3 of 3 Updated April 7, 2011

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 . Then graph the function. Solution: Vertex: Axis of Symmetry: To find -intercepts, solve .

. -intercepts are at and . To find -intercept, . -intercept is at .

, parabola opens up