The Parabola: Complete SAT Guide
Master every aspect of parabolas — from vertex form to optimization problems
1. What Is a Parabola?
A parabola is a U-shaped curve that is the graph of a quadratic function. It is a symmetric curve with a special point called the vertex.
Quadratic Function
A function of the form $f(x) = ax^2 + bx + c$ where $a \neq 0$. It is a degree-2 polynomial — the highest power of $x$ is 2.
Why Parabolas Matter:
- Projectile motion — path of a thrown ball
- Optimization — maximizing area, revenue, profit
- SAT favorite — appears in 3–6 questions per test
Key Features of a Parabola
| Feature | Description | How to Find |
|---|---|---|
| Vertex | Turning point (highest or lowest) | $(h, k)$ or $\left(-\frac{b}{2a},\; f\!\left(-\frac{b}{2a}\right)\right)$ |
| Axis of Symmetry | Vertical line through vertex | $x = h$ or $x = -\frac{b}{2a}$ |
| Direction | Opens up or down | $a > 0$: up | $a < 0$: down |
| y-intercept | Where graph crosses y-axis | $(0, c)$ |
| x-intercepts | Where graph crosses x-axis (roots/zeros) | Set $y = 0$, solve |
| Domain | All possible x-values | Always $(-\infty, \infty)$ |
| Range | All possible y-values | $[k, \infty)$ or $(-\infty, k]$ |
2. Three Forms of Quadratic Equations
The same parabola can be written in three different forms. Each form reveals different information at a glance.
Standard Form
$y = ax^2 + bx + c$
Reveals: y-intercept directly ($c$)
Best for: Finding discriminant, using quadratic formula
y-intercept = $(0, 3)$
Vertex Form
$y = a(x - h)^2 + k$
Reveals: vertex directly at $(h, k)$
Best for: Graphing, finding max/min
Vertex = $(2, -1)$
Factored Form
$y = a(x - r_1)(x - r_2)$
Reveals: x-intercepts directly ($r_1$ and $r_2$)
Best for: Finding roots/zeros
x-intercepts = $(1,0)$ and $(3,0)$
Same Parabola, Three Forms
$y = x^2 - 4x + 3 \;=\; (x-2)^2 - 1 \;=\; (x-1)(x-3)$
All three describe the exact same curve!