Quadratic Functions and Their Graphs

Introduction to Quadratic Functions

Quadratic functions are fundamental in precalculus and are represented by equations of the form , where , , and are real numbers and . The graph of a quadratic function is a parabola, which can open upward or downward depending on the sign of .

• Vertex: The point where the parabola changes direction.

• Axis of Symmetry: The vertical line that passes through the vertex and divides the parabola into two symmetric halves.

• Maximum/Minimum Value: The highest or lowest point on the graph, corresponding to the -value of the vertex.

Graphing Quadratic Functions in Vertex Form

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

• Vertex:

• Axis of Symmetry:

• Direction: If , the parabola opens upward (minimum value). If , it opens downward (maximum value).

Example 1

Given , the vertex is and the axis of symmetry is . The minimum value is .

Example 2

Given , the vertex is and the axis of symmetry is . The maximum value is $3$.

Finding the Vertex, Axis of Symmetry, and Extrema from Standard Form

Standard Form and Completing the Square

Quadratic functions can also be written in standard form: . To find the vertex, we use:

• x-coordinate of vertex:

• y-coordinate of vertex: Substitute into

Example

For :

• , ,

• Vertex:

• Axis of Symmetry:

• Minimum Value:

Maximum and Minimum Values

Determining Extrema

The maximum or minimum value of occurs at the vertex. The nature of the extremum depends on the sign of :

• If : Parabola opens upward; vertex is a minimum.

• If : Parabola opens downward; vertex is a maximum.

Applied Problems Involving Quadratic Functions

Projectile Motion Example

Quadratic functions model real-world phenomena such as projectile motion. For example, the height of a stone thrown upward is given by:

• Maximum Height: Occurs at

• Application: Substitute into to find the maximum height.

Area Optimization Example

Quadratic functions can be used to maximize area, such as finding the dimensions of a rectangular pond with a fixed perimeter.

• Set up a quadratic equation for area in terms of one variable.

• Find the vertex to determine the maximum area.

Summary Table: Properties of Quadratic Functions

Key Formulas

• Vertex (Standard Form):

• Vertex Form:

• Axis of Symmetry: (vertex form), (standard form)

Additional info:

• TI-84 or similar calculators can be used to graph quadratic functions and find vertex form efficiently.

• Quadratic functions are essential for modeling optimization and motion problems in mathematics and science.