Polynomial and Rational Functions

Quadratic Functions

Quadratic functions are a fundamental topic in College Algebra, characterized by their parabolic graphs and applications in optimization problems. This section covers the standard forms, graphing techniques, and methods for finding minimum and maximum values.

Recognizing Characteristics of Parabolas

• Definition: A quadratic function is any function of the form or , where .

• Graph: The graph of a quadratic function is a parabola.

• Vertex: The vertex is the point in the standard form, or in the general form.

• Axis of Symmetry: The line or divides the parabola into two symmetric halves.

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

Graphing Quadratic Functions in Standard Form

To graph a quadratic function in standard form :

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

2. Identify the vertex .

3. Find 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: For :

• (parabola opens downward)

• Vertex:

• x-intercepts: and

• y-intercept:

Graphing Quadratic Functions in General Form

For :

1. Direction: (upward), (downward)

2. Vertex: ,

3. x-intercepts: Solve

4. y-intercept:

5. Plot and connect points

Example: For :

• , ,

• Vertex: , (vertex at )

• x-intercepts: and

• y-intercept:

• Axis of symmetry:

Vertex of a Parabola

• Vertex Formula: For , the vertex is at .

• Substitute into to find the y-coordinate.

Minimum and Maximum Values

• If , the function has a minimum at , value .

• If , the function has a maximum at , value .

Example: For :

• (minimum)

• Vertex: ,

• Domain: All real numbers

• Range:

Strategy for Maximizing or Minimizing Quadratic Functions

1. Identify the quantity to maximize or minimize.

2. Express it as a function in one variable.

3. Rewrite in standard quadratic form.

4. Calculate for the extremum.

5. Answer the problem's question.

Application Example: Maximizing Area

Problem: Maximize the area of a rectangle with 120 feet of fencing.

• Let and be the rectangle's dimensions.

• Perimeter:

• Area:

• Maximum area at

• Dimensions: feet

• Maximum area: $900$ square feet