Quadratic Functions and Their Graphs: Standard, Vertex, and Factored Forms — SAT Math

Introduction

Quadratic functions appear throughout the SAT's Advanced Math domain (≈35% of the section). A signature question type gives you a parabola's equation in one form and asks about a feature — its vertex, axis of symmetry, x-intercepts, y-intercept, or maximum/minimum value — that is hardest to read from that form. The fastest strategy: recognize which form reveals each feature without extra algebra, and convert when needed.

This note focuses on the three standard algebraic forms and the key features each one displays. For quadratics involving a transformation of a parent function, see Function Notation, Composition, and Transformations. Solving for the roots algebraically is covered in Solving Quadratic Equations.

Core Concept

Every quadratic $f(x) = ax^2 + bx + c$ with $a \neq 0$ graphs as a parabola. The same parabola can be written in three equivalent forms, each spotlighting different features.

The Three Forms at a Glance

Form	Equation	Feature revealed directly
Standard	$f(x) = ax^2 + bx + c$	y-intercept $(0,\, c)$
Vertex	$f(x) = a(x-h)^2 + k$	Vertex $(h,\, k)$ ; axis $x = h$ ; max/min value $k$
Factored	$f(x) = a(x-r_1)(x-r_2)$	x-intercepts (zeros) $x = r_1$ and $x = r_2$

The coefficient $a$ is the same in all three forms and tells you the direction of opening:

$a > 0$ : opens upward → minimum at the vertex
$a < 0$ : opens downward → maximum at the vertex

Axis of Symmetry

The axis of symmetry always passes through the vertex. It can be found from any form:

From vertex form: $x = h$ (read directly)
From factored form: $x = \dfrac{r_1 + r_2}{2}$ (midpoint of the two zeros)
From standard form: $x = -\dfrac{b}{2a}$ (must be memorized; not on the reference sheet)

Quick Illustration

Let $f(x) = 2x^2 - 12x + 10$ . All three forms represent the same parabola:

\underbrace{2x^2 - 12x + 10}_{\text{standard: y-int} = 10} \;=\; \underbrace{2(x-1)(x-5)}_{\text{factored: zeros at }1,5} \;=\; \underbrace{2(x-3)^2 - 8}_{\text{vertex: min}=-8 \text{ at } x=3}

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