Quadratic Functions

A quadratic function is a type of polynomial function of the form:

f(x) = ax² + bx + c

where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic function is a parabola, which is a U-shaped curve.

Key Concepts

1. Vertex: The vertex of a parabola is the highest or lowest point on the graph. The vertex form of a quadratic function is:

f(x) = a(x - h)² + k

2. Axis of Symmetry: The line that divides the parabola into two symmetrical halves.
3. Zeroes/Roots: The values of x for which f(x) = 0. These are the x-intercepts of the graph.
4. Maximum/Minimum: The highest or lowest point on the graph, which is the vertex of the parabola.

Key Formulas

• Vertex Form: f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.
• Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a), used to find the x-intercepts of the parabola.
• Discriminant: The expression b² - 4ac inside the quadratic formula, which determines the nature of the roots (real, imaginary, or equal).

Graphing Quadratic Functions

To graph a quadratic function, you can follow these steps:

1. Identify the vertex (h, k) using the formula f(x) = a(x - h)² + k.
2. Find the x-intercepts by solving the equation f(x) = 0 using the quadratic formula or factoring.
3. Find the axis of symmetry, which is the line x = h.
4. Plot the vertex, x-intercepts, and axis of symmetry, then sketch the parabola.

Applications of Quadratic Functions

Quadratic functions are used to model various real-world situations, such as projectile motion, engineering designs, and financial analysis. Understanding quadratic functions helps in solving optimization problems and predicting the behavior of physical systems.

Practice Problems

Try solving the following problems to reinforce your understanding of quadratic functions:

1. Find the vertex, axis of symmetry, and x-intercepts of the function f(x) = 2x² - 4x - 6.
2. Graph the function f(x) = -x² + 4x - 3 and determine the maximum or minimum point.
3. Solve the equation 3x² + 2x - 5 = 0 using the quadratic formula.

Hope this study guide helps you in understanding quadratic functions better!