Comprehensive Study Guide for Absolute Value and Exponential Functions

Standard Form

A quadratic function is represented in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

Graph Shape

The graph of a quadratic function is a parabola.

Direction of Opening

If $a > 0$, the parabola opens upwards. If $a < 0$, it opens downwards.

Vertex

The vertex is the highest or lowest point of the parabola and can be found using the formula $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.

Axis of Symmetry

The axis of symmetry for the parabola is given by the equation $x = -\frac{b}{2a}$.

X-Intercepts

Solutions to the equation $f(x) = 0$, typically found using factoring, completing the square, or the quadratic formula.

Y-Intercept

The y-intercept occurs where $x = 0$, calculated as $f(0) = c$.

Table of Values

A set of selected $x$ values used to plot points for graphing the quadratic function.

Example Function

An example quadratic function is $f(x) = x^2 - 4x + 3$.

Real-World Applications