Quadratic Functions

Definition and Properties

A quadratic function is a polynomial function of degree 2, generally written as , where , , and are real numbers and . The graph of a quadratic function is called a parabola.

• Domain:

• Vertex: The turning point of the parabola, given by

• Axis of Symmetry: The vertical line

• Y-intercept: The point

• Opening Direction: Upward if , downward if

• Stretch/Compression: The greater , the narrower (more "vertically stretched") the parabola

• Graph of : Parabola with vertex at the origin, symmetric to the -axis

Example:

• Graph and on the same axes. The first opens upward, the second downward.

Standard Form and Vertex Form

The standard form of a quadratic function is , where is the vertex.

• Allows easy identification of the vertex and axis of symmetry.

• Can be converted from general form by completing the square.

Vertex Formula:

• Given , the vertex is:

k = f(h)

Quadratic Formula:

To find -intercepts (roots), solve using:

Example:

• Find the vertex and intercepts of :

• Vertex:

• Axis:

• Y-intercept:

Applications of Quadratic Functions

• Maximum or minimum values occur at the vertex.

• Used in optimization problems, such as maximizing revenue or minimizing cost.

Example:

• Given , find the price that maximizes revenue:

Maximum revenue:

Polynomial Functions

Definition and Classification

A polynomial function of degree has the form , where and is a nonnegative integer.

• Leading term:

• Constant term:

• Leading coefficient:

Types of Polynomial Functions:

• Constant functions:

• Linear functions:

• Quadratic functions:

• Cubic functions:

General Properties:

• Domain:

• Continuous graphs (no breaks or jumps)

• Smooth graphs (no sharp corners)

End Behavior and Leading Term Test

The Leading Term Test describes the end behavior of polynomial functions:

• If is even and : Rises left and right

• If is even and : Falls left and right

• If is odd and : Falls left, rises right

• If is odd and : Rises left, falls right

Example:

• : Degree 4, leading coefficient -3. Falls left and right.

• : Degree 5, leading coefficient 2. Falls left, rises right.

Zeros and Multiplicity

A zero of a polynomial function is a solution to . The multiplicity of a zero is the number of times a factor occurs in the factorization of .

• If multiplicity is odd, the graph crosses the -axis at the zero.

• If multiplicity is even, the graph touches but does not cross the -axis at the zero.

• Higher multiplicity makes the graph flatter near the zero.

Example Table:

Example:

• For , zeros are 0 (multiplicity 2), 2 (multiplicity 1), -5 (multiplicity 2).

Polynomial Division

Long Division and Synthetic Division

Polynomial division is used to factor polynomials and find their zeros. Two main methods are long division and synthetic division.

• Long division: Similar to numerical long division, used for dividing polynomials by other polynomials.

• Synthetic division: A shortcut method for dividing by linear factors of the form .

Division Algorithm:

If is divided by , then , where is the quotient and is the remainder, with degree of less than degree of .

Example Table:

Example:

• Divide by using both methods.

Summary Table: Types of Polynomial Functions

Additional info: These notes are based on "Precalculus: A Right Triangle Approach, 4th ed." by Ratti, McWaters, and Skrzpek, and cover foundational concepts in quadratic and polynomial functions, including graphing, properties, and division techniques.