The Quadratic Function | Moosa Academy

The Quadratic Function — Moosa Academy

A function is called quadratic because the highest power of its variable is 2 — the variable is squared. That single feature forces its graph into one shape, the parabola, and everything else about the function follows from the three coefficients that describe it.

Concept Standard form

f(x) = ax^{2} + bx + c, \qquad a \neq 0

a — sets the opening direction and the width.

b — shifts the axis of symmetry sideways.

c — the y-intercept, since

f(0) = c

The condition $a \neq 0$ is not a technicality. With $a = 0$ the squared term vanishes and the function collapses to $f(x) = bx + c$ , a straight line.

Concept The vertex and axis of symmetry

x = -\frac{b}{2a}

This vertical line is the axis of symmetry; the parabola is a mirror image across it. The vertex sits on it, and its height comes from substituting that x back into the function.

a > 0

the vertex is the lowest point — the minimum value.

a < 0

the vertex is the highest point — the maximum value.

b = 0

the axis is the y-axis itself, and the vertex lies at

(0,\ c)

Note the minus sign: a positive $b$ moves the axis to the left, a negative $b$ to the right — the opposite of what the sign suggests.

Note Roots and the discriminant

The roots are the solutions of $f(x) = 0$ — where the parabola meets the x-axis. A quadratic always has two, though they may coincide or fail to be real. The discriminant decides which:

Δ = b² − 4ac	Roots	Graph
Δ > 0	Two distinct real	Crosses the axis twice
Δ = 0	One repeated	Touches the axis once
Δ < 0	Two non-real	Never meets the axis

x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Example Analysing

f(x) = x^{2} - 4x + 3

a = 1

b = -4

c = 3

. Since

a > 0

the parabola opens upward.

Axis:

x = -\dfrac{-4}{2(1)} = 2

Vertex height:

f(2) = 4 - 8 + 3 = -1

Discriminant:

\Delta = 16 - 12 = 4 > 0

, so two distinct real roots.

x = \dfrac{4 \pm 2}{2}

, giving

x = 3

and

x = 1

⟹ vertex (2, −1), roots x = 1 and x = 3

The vertex lies below the axis and the curve opens upward, so it must cross twice — the discriminant only confirmed what the shape already implied.

Note Where quadratics come from

Anything that rises and then falls — a thrown ball, a jet of water in a fountain, the path of a projectile — traces a parabola, because constant acceleration makes displacement depend on the square of time. The same shape appears in reverse for objects that fall and rebound, and in optimisation problems where a maximum or minimum is wanted.

Summary

A quadratic is f(x) = ax² + bx + c with a ≠ 0; its graph is a parabola.
a sets the direction, b shifts the axis of symmetry, c is the y-intercept.
The axis of symmetry is x = −b/2a, and the vertex sits on it.
The discriminant b² − 4ac decides whether there are two, one or no real roots.