What is a Quadratic Function?

Quadratic Function

A function of the form $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq = 0$ . The highest power of the variable is 2.

Quadratic functions are one of the most powerful tools in mathematical modelling. Unlike linear functions (which give straight lines), quadratic functions produce a curved shape called a parabola. This curve appears everywhere in the real world — from the path of a thrown ball to the shape of a satellite dish.

The three coefficients each play a distinct role:

$a$ controls the width and direction of the parabola. If $a > 0$ , it opens upward (like a bowl); if $a < 0$ , it opens downward (like an arch).
$b$ influences the position of the axis of symmetry.
$c$ is the y-intercept — the value of the function when $x = 0$ .

Analogy

Think of throwing a ball into the air. It rises, slows down, reaches a peak, then falls back down. That arc is a parabola — a quadratic function in motion! The ball's height at any time $t$ can be modelled by a quadratic equation.

What is a Quadratic Function?

Key Features of a Parabola

To use quadratic functions as models, you need to read and interpret their graphs confidently. Every parabola has several key features:

Vertex

The turning point of the parabola — either the maximum (highest) or minimum (lowest) point. Its coordinates are $(- \frac{b}{2 a}, f! (- \frac{b}{2 a}))$ .

Axis of Symmetry

The vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is $x = - \frac{b}{2 a}$ .

Roots (x-intercepts)

The values of $x$ where the parabola crosses the x-axis, i.e., where $f (x) = 0$ . A quadratic can have 0, 1, or 2 real roots.

y-intercept

The point where the parabola crosses the y-axis. Found by substituting $x = 0$ , giving the point $(0, c)$ .

Here is a summary of these features:

Feature	Formula / Location
Vertex	$x = - \frac{b}{2 a}$ , then find $y$
Axis of symmetry	$x = - \frac{b}{2 a}$
y-intercept	$(0, c)$
Roots	Solve $a x^{2} + b x + c = 0$

Key Features of a Parabola

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