SAT Advanced Math: Quadratic Functions and the Parabola

Three forms of the quadratic, the discriminant rule, when to factor vs. use the formula, and the Vieta shortcut that solves "sum of the roots" questions in under 10 seconds.

9 min TEKS ADV SAT Math

Quadratics are the most-tested topic in the Advanced Math domain — expect 4–6 questions on parabolas, the quadratic formula, factoring, and vertex form. Three forms of the same equation, three different uses.

The three forms of a quadratic

Each form reveals a different feature

Standard: y = ax² + bx + c → reveals the y-intercept (c)
Vertex: y = a(x − h)² + k → reveals the vertex (h, k)
Factored: y = a(x − p)(x − q) → reveals the roots (p and q)

The vertex sits exactly halfway between the two roots. The axis of symmetry x = h passes through it.

The quadratic formula

For any quadratic ax² + bx + c = 0, the roots are given by:

x = (−b ± √(b² − 4ac)) / (2a) Memorize this. The Bluebook reference sheet does NOT include it.

When to factor vs. when to use the formula

Always try to factor first — it's faster when the numbers are clean. Use the quadratic formula when factoring fails or coefficients are messy.

x² − 5x + 6 = 0 (x − 2)(x − 3) = 0 x = 2 or x = 3 Need two numbers that multiply to +6 and add to −5: −2 and −3.

2x² + 3x − 5 = 0 x = (−3 ± √(9 + 40)) / 4 x = (−3 ± 7) / 4 x = 1 or x = −5/2 Discriminant 49 → perfect square → clean roots.

The discriminant tells you how many real roots

Discriminant: D = b² − 4ac D > 0 → two real roots (parabola crosses x-axis twice) D = 0 → one real root (parabola just touches x-axis) D < 0 → no real roots (parabola never crosses) SAT loves asking "for what value of k does this have exactly one solution?" — set D = 0.

Vertex form and completing the square

If you need the vertex from standard form ax² + bx + c, use the shortcut:

h = −b / (2a) k = f(h) — plug h back in Example: y = x² − 6x + 11 → h = 3, k = 9 − 18 + 11 = 2 → vertex (3, 2)

Desmos for quadratics

Type the quadratic and Desmos shows the parabola with clickable roots, vertex, and y-intercept. For "find the sum of the roots" questions, this is a 10-second solve.

Vieta's shortcut

For ax² + bx + c = 0: sum of roots = −b/a and product of roots = c/a. Skip the entire solve when only the sum or product is asked.

Which way does the parabola open?

The sign of a in y = ax² + bx + c controls the direction.

a > 0 → parabola opens upward, vertex is a minimum
a < 0 → parabola opens downward, vertex is a maximum
Larger |a| → narrower parabola; smaller |a| → wider

Word problem: projectile motion

A ball's height (in feet) at time t (in seconds) is h(t) = −16t² + 48t + 6. What is the maximum height? a = −16, b = 48 → t = −b/(2a) = −48/(−32) = 1.5 sec h(1.5) = −16(2.25) + 48(1.5) + 6 = −36 + 72 + 6 = 42 ft Projectile problems are quadratics in disguise. Always find the vertex's t-value first.

Function transformations

For y = a(x − h)² + k, each parameter shifts or scales the basic parabola y = x²:

h shifts horizontally — right by h (note the minus sign inside)
k shifts vertically — up by k
a scales vertically — stretches if |a| > 1, compresses if |a| < 1, flips if negative

Common mistakes

Forgetting the second root — quadratics have two solutions unless D = 0
Sign error in the quadratic formula (the −b in front)
Confusing vertex form coordinates: y = a(x − 3)² + 2 has vertex (3, 2), not (−3, 2)
Using the discriminant formula but writing 4ac instead of b² − 4ac
Assuming the parabola opens upward when a is negative

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