Quadratic Functions and Parabolas

Introduction to Quadratic Functions

Quadratic functions are polynomial functions of degree two, commonly written in the form , where , , and are constants and . The graph of a quadratic function is a parabola, which may open upwards or downwards depending on the sign of .

• Standard Form:

• Vertex Form: , where is the vertex of the parabola

• Axis of Symmetry: The vertical line

Graphing Quadratic Functions

To graph a quadratic function, identify key features such as the vertex, axis of symmetry, intercepts, and direction of opening.

• Vertex: The highest or lowest point of the parabola, given by in standard form

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

• X-intercepts: The points where the graph crosses the x-axis, found by solving

• Direction: If , the parabola opens upward; if , it opens downward

Example Table of Values

Tables of values help plot points for the graph:

Vertex and Intercepts

The vertex and intercepts are crucial for understanding the shape and position of the parabola.

• Vertex Example: means the vertex is at

• X-intercepts: Points where , e.g., and

• Y-intercept: Point where , e.g.,

Domain and Range

The domain and range describe the set of possible input and output values for the function.

Quadratic Function Transformations

Transformations affect the position and shape of the parabola.

• Vertical Stretch/Compression: The value of changes the width

• Horizontal/Vertical Shifts: Changing and in vertex form moves the parabola

• Reflection: Negative reflects the parabola over the x-axis

Example Functions and Vertices

• has vertex

• has vertex

Applications and Problem Solving

Quadratic functions are used to model real-world phenomena such as projectile motion, optimization problems, and economics.

• Finding Intercepts: Set for x-intercepts, for y-intercept

• Vertex Calculation: Use for standard form

• Domain and Range: Analyze the graph's direction and vertex

Summary Table: Key Properties of Quadratic Functions

Additional info: Some values and points were inferred from context and standard quadratic function properties.