Quadratic Function

A quadratic function is a type of polynomial function that can be written in 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 can open upwards or downwards depending on the value of a.

Key Concepts

1. Vertex: The vertex of a parabola is the highest or lowest point on the graph. It occurs at the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate.
2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetric halves.
3. Y-Intercept: The y-intercept is the point where the graph intersects the y-axis. It occurs at the point (0, c).
4. X-Intercepts (Zeros/Roots): The x-intercepts are the points where the graph intersects the x-axis. They represent the solutions to the quadratic equation ax² + bx + c = 0.
5. Vertex Form: The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Graphing Quadratic Functions

To graph a quadratic function, follow these steps:

1. Identify the vertex (h, k) and plot it on the coordinate plane.
2. Find the y-intercept at (0, c).
3. Determine the axis of symmetry, which is the line x = h.
4. Calculate the x-intercepts by solving ax² + bx + c = 0 using the quadratic formula or factoring.
5. Use additional points or the symmetry of the parabola to complete the graph.

Quadratic Formula

The quadratic formula is used to solve for the x-intercepts of a quadratic function. It is given by:

x = ^{-b ± √(b2 - 4ac)} / 2a

Study Tips

• Practice identifying the key components of a quadratic function, such as the vertex, axis of symmetry, and intercepts.
• Learn to manipulate quadratic functions into vertex form to easily identify the vertex and other properties.
• Master the quadratic formula and different methods for solving quadratic equations, including factoring and completing the square.
• Utilize graphing calculators or software to visualize and explore various quadratic functions and their properties.
• Work through plenty of practice problems to reinforce your understanding of quadratic functions and their applications.

Understanding quadratic functions is essential for various mathematical and real-world applications, including physics, engineering, and finance. Mastering this topic will provide a solid foundation for further studies in algebra and calculus.