Quadratic Function

The quadratic function is one of the most important functions in algebra. Its graph — the parabola — appears everywhere, from projectile motion to optimization problems.

Standard Form

📐Definition — Quadratic Function

A quadratic function has the form:

$f(x) = ax^2 + bx + c, \quad a \neq 0$

If $a > 0$ , the parabola opens upward.
If $a < 0$ , the parabola opens downward.
The larger $|a|$ is, the narrower the parabola.

Axis of Symmetry and Vertex

⚡Theorem — Vertex Coordinates

For $f(x) = ax^2 + bx + c$ , the axis of symmetry is:

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

The vertex is:

$\left(-\frac{b}{2a},\; c - \frac{b^2}{4a}\right)$

The vertex gives the global minimum if $a > 0$ , or the global maximum if $a < 0$ .

Vertex Form

The vertex form is:

$f(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex. To convert from standard form, complete the square.

✎Example — Completing the Square

Convert $f(x) = x^2 - 6x + 5$ to vertex form.

Solution.

$f(x) = x^2 - 6x + 5 = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4$

The vertex is $(3, -4)$ . The axis of symmetry is $x = 3$ .

f(x) = x² + +

a = 1.0

b = 0.0

c = 0.0

D = b²-4ac = 0.00Vertex: (0.00, 0.00)Roots: 0.00

The Discriminant and $x$ -intercepts

The $x$ -intercepts are the solutions of $ax^2 + bx + c = 0$ . Their existence depends on the discriminant:

$D = b^2 - 4ac$

Discriminant	Roots	Graph
$D > 0$	Two distinct: $x = \dfrac{-b \pm \sqrt{D}}{2a}$	Crosses $x$ -axis twice
$D = 0$	One repeated: $x = -\dfrac{b}{2a}$	Touches $x$ -axis at vertex
$D < 0$	No real roots	Does not cross $x$ -axis

Worked Example

✎Example — Complete Analysis

Find the vertex, axis of symmetry, and $x$ -intercepts of $f(x) = 2x^2 - 4x - 6$ .

Solution.

Vertex: $x = -\dfrac{-4}{2 \cdot 2} = 1$ , $\;f(1) = 2 - 4 - 6 = -8$ . Vertex: $(1, -8)$ .

Axis of symmetry: $x = 1$ .

Discriminant: $D = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 > 0$ .

Roots: $x = \dfrac{4 \pm 8}{4}$ , giving $x = 3$ and $x = -1$ .

The parabola opens upward ( $a = 2 > 0$ ), has a minimum of $-8$ at $x = 1$ , and crosses the $x$ -axis at $(-1, 0)$ and $(3, 0)$ .

✏Exercise

Find the vertex form and all intercepts of $f(x) = -x^2 + 2x + 3$ .