Polynomial and Quadratic Functions

Polynomial Functions

Polynomial functions are a fundamental class of functions in algebra and precalculus. A polynomial function of degree n is defined as:

• Definition: , where and all coefficients are real numbers.

• Degree: The highest power of x in the function.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Classification: Polynomials are classified by degree: constant (0), linear (1), quadratic (2), cubic (3), quartic (4), etc.

Quadratic Functions

A quadratic function is a specific type of polynomial function of degree 2. Its general form is:

• Definition: , where .

• Simplest Quadratic: is the basic quadratic function.

• Domain:

• Range: for

Parabolas and Their Properties

The graph of a quadratic function is called a parabola. Parabolas have several important properties:

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

• Vertex: The point where the axis of symmetry intersects the parabola; it is the maximum or minimum point.

• Opening Direction: Parabolas open upward if and downward if .

Graphing Quadratic Functions

Graphing Techniques

Quadratic functions can be graphed using transformations and by completing the square. The vertex form is especially useful:

• Vertex Form:

• Vertex:

• Axis of Symmetry:

• Transformations:

  • Opens up if , down if

  • Vertical stretch/shrink depending on

  • Horizontal shift by units

  • Vertical shift by units

Example: Graphing Quadratic Functions

Graphing by plotting points and identifying the vertex and axis of symmetry:

• Vertex:

• Axis of Symmetry:

• Domain:

• Range:

Completing the Square

Completing the square is a method used to rewrite a quadratic function in vertex form, which makes graphing and analysis easier.

• Process: Rewrite as by completing the square.

• Vertex: The resulting gives the vertex.

• Intervals of Increase/Decrease: The function decreases to the vertex and increases after.

Graphing When a ≠ 1

When the leading coefficient , factor from the quadratic terms before completing the square. This affects the width and direction of the parabola.

• Example:

• Vertex:

• Intercepts: Find y-intercept by evaluating ; x-intercepts by solving .

Vertex Formula and Quadratic Models

Vertex Formula

The vertex of a parabola given by can be found using:

• Axis of Symmetry:

• Vertex:

Quadratic Models in Applications

Quadratic functions are used to model real-world phenomena, such as projectile motion and population growth.

• Projectile Motion: models the height of an object projected upward.

• Maximum Height: Occurs at the vertex.

• Interval Analysis: Quadratic inequalities can determine intervals where the function exceeds a certain value.

• Time to Hit Ground: Solve for using the quadratic formula.

Quadratic Regression and Data Modeling

Modeling Data with Quadratic Functions

Quadratic regression is used to fit a quadratic model to real-world data, such as hospital outpatient visits over time.

• Scatter Diagram: Visual representation of data points.

• Quadratic Regression: Finds the best-fit quadratic equation .

• Prediction: Use the model to estimate values for future years.

Solving Quadratic Equations

Quadratic Formula and X-Intercepts

The quadratic formula is used to find the x-intercepts (roots) of a quadratic function:

• Quadratic Formula:

• Discriminant: determines the nature of the roots:

  • If , two real roots

  • If , one real root

  • If , no real roots

Summary Table: Properties of Quadratic Functions

Additional info: Academic context was added to clarify the process of completing the square, the use of quadratic regression, and the application of quadratic models in projectile motion and data prediction.