Polynomial and Rational Functions

Quadratic Functions

A quadratic function is a polynomial function of degree 2, typically written in the form , where a, b, and c are real numbers and . The graph of a quadratic function is a parabola.

• Standard Form: where is the vertex.

• General Form:

• Axis of Symmetry: The line (in standard form) or (in general form).

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

Example: is a quadratic function. Its graph is a parabola.

Graphing Quadratic Functions in Standard Form

To graph a quadratic function in standard form :

• Step 1: Identify , , and .

• Step 2: Determine the direction of opening (up if , down if ).

• Step 3: Find the vertex . The vertex is the minimum (if ) or maximum (if ).

• Step 4: Find the x-intercepts by solving .

• Step 5: Find the y-intercept by evaluating .

• Step 6: Sketch the graph, plot the points, and draw the axis of symmetry.

Graphing Quadratic Functions in General Form

For :

• Step 1: Identify , , and .

• Step 2: Determine the direction of opening.

• Step 3: Find the vertex using and .

• Step 4: Find the x-intercepts by solving .

• Step 5: Find the y-intercept ().

• Step 6: Use symmetry about the axis to find additional points.

• Step 7: Draw the parabola through the points found.

Example: Graphing a Quadratic Function

Consider .

• Vertex: ,

• Axis of Symmetry:

• Minimum Value: at

• Y-intercept:

• X-intercepts: Solve

Applications: Maximum Area Problem

Quadratic functions are often used to model real-world problems involving maximum or minimum values. For example, maximizing the area of a rectangular pen with a fixed perimeter.

• Problem: A farmer has 100 feet of fencing and uses a stone wall for one side. The other three sides use the fencing.

• Let: = length of each of the two sides perpendicular to the wall, = length parallel to the wall.

• Constraint:

• Area:

• Maximum Area: The area function is quadratic and opens downward (), so the maximum occurs at the vertex.

• Vertex:

• Dimensions: ft, ft

• Maximum Area: ft2

Solving Quadratic Inequalities

Quadratic functions can also be used to solve inequalities, such as . The solution involves finding the values of for which the function is positive.

• Step 1: Solve to find critical points.

• Step 2: Test intervals between and outside the roots to determine where the function is greater than zero.

• Step 3: Express the solution as an interval.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers, . The range depends on the direction of the parabola:

• If : Range is where is the minimum value at the vertex.

• If : Range is where is the maximum value at the vertex.