Quadratic Functions
A quadratic function is a polynomial function of degree 2, typically expressed in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. The graph of a quadratic function is called a parabola, which is a symmetric, U-shaped curve.
Standard Form
The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. In this form:
- $a$ determines the direction and steepness of the parabola (for $a>0$ , the graph is concave up, and for $a<0$ is concave down
- $b$ influences the axis of symmetry
- $c$ is the y-intercept
Consider the function $f(x) = 2x^2 - 4x + 3$. Here, $a=2$, $b=-4$, and $c=3$.
Y-intercept
The y-intercept of a quadratic function is the point where the parabola crosses the y-axis. It occurs when $x = 0$, and its coordinates are always (0, c).
ExampleFor $f(x) = 2x^2 - 4x + 3$, the y-intercept is (0, 3).
Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is given by:
$$ x = -\frac{b}{2a} $$
ExampleFor $f(x) = 2x^2 - 4x + 3$, the axis of symmetry is $x = -\frac{-4}{2(2)} = 1$.
