Quadratic Equations and Functions

Converting to Vertex Form

Quadratic functions can be expressed in several forms, with the vertex form being particularly useful for identifying the vertex of the parabola. The process of converting from standard form to vertex form allows for easier analysis and application of quadratic functions.

• Standard Form:

• Vertex Form: , where is the vertex and

• Finding the Vertex: The x-coordinate of the vertex is

• Substitute into to find the y-coordinate .

Example 1: Convert the following to vertex form:

• a)

• b)

Solution Outline:

1. Find

2. Compute

3. Write in vertex form:

Example: For :

• ,

• Vertex form:

Applications of Quadratic Functions

Quadratic functions are widely used in real-world applications, especially in optimization problems such as maximizing revenue or area.

• Revenue Maximization: Revenue is the product of price and quantity sold.

• Area Optimization: Maximizing area with a fixed perimeter is a classic quadratic application.

Example 2: A store sells t-shirts at price . Find the price that maximizes revenue.

• Revenue function:

• Maximum at

• Maximum revenue occurs at

Example 3: Fencing a rectangular corral with 1200 ft of fence, one side along a creek (no fence needed there). Maximize area.

• Let width = , length =

• Perimeter:

• Area:

• Express in terms of :

• Area as a function of :

• Find that maximizes using vertex formula

Example 4: Hockey team ticket pricing and attendance:

• Let = ticket price, = attendance

• At ,

• For each decrease, increases by 1000:

• Revenue:

• Expand and find that maximizes using vertex formula

• Find where for no revenue

Quadratic Expressions and Factoring

Expanding and Factoring Quadratic Expressions

Quadratic expressions can be expanded or factored to reveal their structure. Factoring is the process of writing a quadratic as a product of two binomials.

• Standard Form:

• Factoring: Writing as where and satisfy , (when )

Example 1: Factor the following:

• a) →

• b) →

• c) →

When : Use grouping or diamond-square methods.

• Grouping: Split the middle term and factor by grouping.

• Diamond-Square: Use a diagram to organize factors.

Example 2: Factor:

• a)

• b)

Special Factoring Cases

Some quadratics are perfect squares or differences of squares, allowing for quick factoring.

• Perfect Square Trinomial:

• Difference of Squares:

Example 3: Factor:

• a)

• b)

• c) (not a difference of squares, but can be written as ; not factorable over real numbers)

Example 4: Expand

Key Points:

• Factoring is essential for solving quadratic equations and simplifying expressions.

• Recognize special forms for efficient factoring.

• Quadratic functions model many real-world scenarios, especially those involving maximization or minimization.

Additional info: The notes reference "Check For Your Understanding" problems and supplemental worksheets, which are practice exercises for mastery.