Quadratic Functions and Parabolas

Three Forms of Quadratic Functions

Form	Equation	Key Info
Standard	$f(x) = ax^2 + bx + c$	y-intercept = $c$
Vertex	$f(x) = a(x-h)^2 + k$	vertex = $(h, k)$
Factored	$f(x) = a(x-r)(x-s)$	x-intercepts = $r, s$

Direction of Opening

• $a > 0$: opens upward (U-shaped) → has a minimum
• $a < 0$: opens downward (∩-shaped) → has a maximum

Vertex

The vertex is the turning point — the minimum or maximum point.

From standard form: $x_v = -\frac{b}{2a}$, then $y_v = f(x_v)$.

From vertex form: vertex is directly $(h, k)$.

Axis of Symmetry

The vertical line through the vertex: $x = -\frac{b}{2a}$ (or $x = h$ in vertex form).

Y-Intercept

Set $x = 0$: $f(0) = c$ (in standard form).

X-Intercepts (Roots/Zeros)

Set $f(x) = 0$ and solve. The number of x-intercepts depends on the discriminant:

• $\Delta > 0$: 2 x-intercepts
• $\Delta = 0$: 1 x-intercept (vertex on x-axis)
• $\Delta < 0$: 0 x-intercepts

The x-intercepts are symmetric about the axis of symmetry.

Width of the Parabola

$|a|$ controls width: larger $|a|$ → narrower parabola; smaller $|a|$ → wider parabola.

Tip 1: Use $x = -\frac{b}{2a}$ for the Vertex

This is faster than completing the square for most SAT problems.

Tip 2: Identify the Form to Find What You Need

Need the vertex? → Vertex form. Need x-intercepts? → Factored form. Need y-intercept? → Standard form.

Tip 3: The Sign of $a$ Tells You Min vs. Max

$a > 0$: minimum at the vertex. $a < 0$: maximum at the vertex.

Tip 4: X-Intercepts Average to the Axis of Symmetry

If the roots are $r$ and $s$, the axis of symmetry is $x = \frac{r+s}{2}$.

Tip 5: Use Desmos to Graph

Type the quadratic into Desmos to instantly see vertex, intercepts, and shape.