Exploring Quadratic Functions: Key Concepts and Graph Insights

Features of Quadratic Functions

A quadratic function follows the standard form f$x$ = ax² + bx + c, where a, b, and c are constants. You can also write it in vertex form as f$x$ = a$x-h$² + k, which immediately tells you the vertex coordinates (h, k).

Every quadratic function creates a parabola with an axis of symmetry - an invisible vertical line that cuts the parabola into mirror images. You can find this line using the formula x = -b/2a. The vertex is the highest or lowest point on the parabola, depending on which way it opens.

Speaking of opening, the value of 'a' determines the parabola's direction. When a > 0, the parabola opens upward (like a cup), and when a < 0, it opens downward (like an upside-down cup). The larger the absolute value of 'a', the narrower the parabola.

Remember This! The vertex form f$x$ = a$x-h$² + k is super helpful for quickly identifying the vertex (h,k) without doing extra calculations. This saves time on tests!

Let's see how this works in examples:

• For f$x$ = 2x² - 4x + 1: The vertex form is f$x$ = 2$x-1$² - 1, so the vertex is at $1, -1$ and the axis of symmetry is x = 1. Since a = 2 is positive, the parabola opens upward.
• For g$x$ = -3x² + 6x - 2: The vertex form is g$x$ = -3$x-1$² + 1, giving a vertex at (1, 1) and axis of symmetry at x = 1. With a = -3 being negative, the parabola opens downward.