Quadratic Functions and Their Properties

Definition of a Quadratic Function

A quadratic function is a function of the form , where , and , , and are real numbers. The graph of a quadratic function is called a parabola.

• Standard Form:

• Vertex Form:

• Axis of Symmetry: The vertical line (in vertex form) or (in standard form)

Key Properties of Quadratic Functions

• The vertex is the highest or lowest point on the graph (maximum or minimum).

• The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images.

• The direction of the parabola (opening up or down) depends on the sign of :

  • If , the parabola opens upward (minimum point).

  • If , the parabola opens downward (maximum point).

• The y-intercept is at .

• The x-intercepts (if any) are found by solving .

Terminology

Formulas for Quadratic Functions

• Axis of Symmetry:

• Vertex:

• Quadratic Formula (for x-intercepts):

Graphing Quadratic Functions

There are several methods to graph quadratic functions, including using transformations, finding the vertex and axis of symmetry, and identifying intercepts.

Option 1: Graphing by Transformations

• Start with the basic graph of .

• Apply transformations (shifts, stretches, reflections) as indicated by the function's form.

• Example: is a vertical stretch by 2, left shift by 1, and down shift by 3.

Option 2: Graphing by Vertex, Axis, and Intercepts

1. Determine the vertex using and .

2. Find the axis of symmetry ().

3. Find the y-intercept ().

4. Find the x-intercepts by solving .

5. Plot the points and sketch the parabola.

Examples

• Example 1: Graph using transformations.

  • Vertex:

  • Axis of symmetry:

  • Vertical stretch by 2, shift left by 1, down by 3

• Example 2: Graph using the vertex and axis of symmetry.

  • Vertex: ,

  • Vertex:

  • Axis of symmetry:

Properties of the Graph of a Quadratic Function

Finding a Quadratic Function Given a Vertex and a Point

• If the vertex and another point are known, use the vertex form and substitute to solve for .

Maximum and Minimum Values

• If , the vertex is the minimum point (parabola opens up).

• If , the vertex is the maximum point (parabola opens down).

• The maximum or minimum value is the y-coordinate of the vertex.

• Example: For , the maximum value is at .