SL 2.6—Quadratic function

Quadratic Function: A polynomial function of degree 2, written in the form $f(x)=a{x}^2+bx+c$, where $a$, $b$, $c\in \mathbb{R}$ and $a\ne 0$. The condition $a\ne 0$ is essential , if $a=0$, the function becomes linear, not quadratic.

The graph of any quadratic function is called a parabola , a smooth, symmetric, U-shaped (or ∩-shaped) curve.

The sign of $a$ tells you the overall shape:

• If $a>0$: the parabola opens upward (concave up, like a cup ∪)
• If $a<0$: the parabola opens downward (concave down, like a cap ∩)

The magnitude of $|a|$ controls the steepness: a larger $|a|$ gives a narrower parabola, while a smaller $|a|$ gives a wider one.

The standard form of a quadratic is $f(x)=a{x}^2+bx+c$. Each coefficient plays a distinct role:

The y-intercept is always at the point $(0,c)$, since substituting $x=0$ gives $f(0)=c$.