Quadratic Functions

Introduction to the Square Function

The square function is one of the most fundamental elementary functions in algebra, defined as . Its graph is a U-shaped curve known as a parabola.

• Standard form: , where are real numbers and .

• All quadratic functions have graphs that are parabolas, which are similar in shape to the graph of .

• Domain: The domain of any quadratic function is all real numbers, .

Example: The graph of opens upward and is symmetric about the y-axis.

Key Features of Quadratic Functions

• Vertex: The highest or lowest point on the graph of a quadratic function. For , the vertex occurs at .

• Axis of Symmetry: The vertical line that passes through the vertex, given by .

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

• Intercepts:

  • y-intercept: The point where the graph crosses the y-axis, found by evaluating .

  • x-intercepts (zeros): The points where the graph crosses the x-axis, found by solving .

The Quadratic Formula

To find the x-intercepts (real solutions) of a quadratic equation , use the quadratic formula:

The solutions are real if .

Forms of Quadratic Functions

• Standard Form:

• Vertex Form: , where is the vertex.

Axis of symmetry: in vertex form.

Minimum/Maximum Value:

• If , the minimum value is at .

• If , the maximum value is at .

Graphing Quadratic Functions

To graph a quadratic function:

1. Identify the vertex using (standard form) or (vertex form).

2. Find the y-intercept by evaluating .

3. Find the x-intercepts by solving (if real solutions exist).

4. Plot the vertex, intercepts, and additional points as needed.

5. Draw the axis of symmetry and sketch the parabola.

Example: For :

• Vertex:

• y-intercept:

• x-intercepts: Solve using the quadratic formula.

Applications of Quadratic Functions

Quadratic functions are widely used in modeling real-world situations, such as projectile motion, profit maximization, and cost minimization.

• Profit and Revenue: Functions like (revenue) and (cost) are often quadratic. The difference gives the profit function.

• Break-even Points: The values of where represent the break-even points.

• Optimization: The vertex of the profit or cost function gives the maximum profit or minimum cost, respectively.

Summary Table: Properties of Quadratic Functions

Practice and Application

• Given a quadratic function, be able to:

  • Find the domain

  • Identify the vertex

  • Determine the axis of symmetry

  • Find x- and y-intercepts

  • Determine the minimum or maximum value

  • Graph the function

  • State intervals of increase and decrease

• Apply quadratic functions to solve real-world problems involving revenue, cost, and profit.

Example Application: Suppose a company's revenue and cost functions are quadratic. To find the break-even point, set and solve for . To maximize profit, find the vertex of the profit function .