Quadratic Functions

Standard Form and Characteristics

A quadratic function is a polynomial function of degree two, typically written in standard form as , where . The graph of a quadratic function is a parabola with the following properties:

• Vertex: The point is the vertex of the parabola.

• Axis of Symmetry: The line is the axis of symmetry.

• Direction: If , the parabola opens upward; if , it opens downward.

Graphing Quadratic Functions in Standard Form

To graph :

1. Determine the direction of opening by the sign of .

2. Identify the vertex .

3. Find the x-intercepts by solving .

4. Find the y-intercept by computing .

5. Plot the vertex, intercepts, and additional points as needed. Connect with a smooth curve.

Example: Graphing

• Direction: ; parabola opens downward.

• Vertex: .

• X-intercepts: Solve :

• Results: and .

• Y-intercept: .

• Graph: The parabola passes through the calculated points.

Quadratic Functions in General Form

The general form is . The vertex can be found using:

• X-coordinate:

• Y-coordinate:

Example:

• X-coordinate of vertex:

• Coordinates of vertex:

• Vertex:

• X-intercepts: Solve using the quadratic formula:

• Results: and

• Y-intercept:

• Axis of symmetry:

• Graph:

Minimum and Maximum Values of Quadratic Functions

For :

• If , the function has a minimum at .

• If , the function has a maximum at .

• The minimum or maximum value is .

Example:

• ; function has a minimum.

• Minimum occurs at .

• Minimum value:

Domain and Range of Quadratic Functions

The domain of any quadratic function is . The range depends on whether the function has a minimum or maximum:

• If , range is .

• If , range is .

Solving Optimization Problems with Quadratic Functions

Quadratic functions are often used to solve optimization problems, such as maximizing area or minimizing cost. The general strategy is:

1. Identify the quantity to maximize or minimize.

2. Express it as a function in one variable.

3. Rewrite in the form .

4. Calculate to find the optimal value.

5. Answer the question posed by the problem.

Example: Maximizing Area with Fixed Perimeter

Given 120 feet of fencing to enclose a rectangular region, find the dimensions that maximize the area.

• Let and be the rectangle's sides. Perimeter: .

• Area: .

• Express in terms of : .

• Area function: .

• Maximum area occurs at .

• Dimensions: $30 ft; maximum area is $900$ sq ft.