A quadratic function is a function of the form where .

Remark: The above form is sometimes called the expanded form of a quadratic function.

Properties of the graph of a quadratic function.

• The graph of a quadratic function is called a parabola.
• The parabola is said to be concave up (opens upward) if .
• The parabola is said to be concave down (opens downward) if .
• The vertex is the highest or lowest point on the parabola.
• The axis of symmetry is a vertical line that contains the vertex.
• The y-intercept is found by evaluating .
• The quadratic function may or may not have x-intercepts.

Vertex: Assume . The vertex .

Where and .

Example1: Find the vertex for each of the following quadratic functions.

a. b. c.

Example2: Find the x-intercepts (if any) for each of the following quadratic functions.

a. b. c

Graphing Quadratic Functions.

Example 3 Let. Do the following:

• Find the vertex
• Axis of symmetry
• y-intercept
• x-intercepts (if any)
• Graph. Label at least three points, including the vertex.
• Domain
• Range
• Increasing
• Decreasing

Graphing Quadratic Functions.

Example 4 Let. Do the following:

• Find the vertex
• Axis of symmetry
• y-intercept
• x-intercepts (if any)
• Graph. Label at least three points, including the vertex.
• Domain
• Range
• Increasing
• Decreasing

Graphing Quadratic Functions.

Example 5 Let. Do the following:

• Find the vertex
• Axis of symmetry
• y-intercept
• x-intercepts (if any)
• Graph. Label at least three points, including the vertex.
• Domain
• Range
• Increasing
• Decreasing