2.3Quadratic Functions

A quadratic function is a function of the form:

Properties of the Graph of a Quadratic Function

Parabola opens up if a > 0; the vertex is a minimum point.Parabola opens down if a < 0; the vertex is a maximum point.

a > 0Opens up

Vertex is lowest point

Axis of symmetry

Graphs of a quadratic function f(x) = ax2 + bx + c

a < 0Opens down

Vertex is highest point

Axis of symmetry

Steps for Graphing a Quadratic Function by Hand

• Determine the vertex.• Determine the axis of symmetry.• Determine the y-intercept, f(0).• Determine how many x-intercepts the graph has. • If there are no x-intercepts determine another

point from the y-intercept using the axis of symmetry.

• Graph.

Without graphing, locate the vertex and find the axis of symmetry of the following parabola. Does it open up or down?

Vertex:

Since -3 < 0 the parabola opens down.

Finding the vertex by completing the square:

10 0 10

15

15

(0,0)

(2,4)

y x= 2

10 0 10

15

15

(0,0)

(2, -12)

10 0 10

15

15

(2, 0)

(4, -12)

10 0 10

15

15

(2, 13)Vertex

Determine whether the graph opens up or down.Find its vertex, axis of symmetry, y-intercept, x-intercept.

x-coordinate of vertex:

Axis of symmetry:

y-coordinate of vertex:

There are two x-intercepts:

5 0

10

10

Vertex: (-3, -13)

(-5.55, 0) (-0.45, 0)

(0, 5)