Quadratics on the SAT: Vertex Form and Question Types

Quadratics on the SAT belong to Advanced Math, the domain that makes up about 35% of the scored Math questions. The single most useful idea is vertex form, f(x) = a(x − h)² + k: it shows the vertex (h, k) directly, and k is the minimum value when a is positive or the maximum when a is negative. The skills College Board lists for quadratics come down to choosing the right form, reading a feature from it, solving, and counting solutions.

This guide shows where quadratics sit in College Board's skill list, what each form of a quadratic tells you, and the question types you will meet, with worked examples written for this article.

Where quadratics sit in SAT Math

College Board's Advanced Math domain covers absolute value, quadratic, exponential, polynomial, rational and radical equations, and it accounts for 13–15 of the scored Math questions. Quadratics run through three of its four skills. College Board's assessment framework lists, among others, these testing points:

• Equivalent expressions: factorising, including a difference of two squares and a trinomial written as the product of two binomials.
• Nonlinear equations in one variable and systems of equations in two variables: solving quadratic equations presented in many forms, deciding whether a quadratic has no real solution, one or two, solving with the quadratic formula or by completing the square when that is the most efficient method, and linking the solutions of a linear-and-quadratic system to the points where the graphs meet.
• Nonlinear functions: building quadratic models, interpreting points and intercepts on a graph in context, connecting tables, equations and graphs (including shifted graphs), and determining the most suitable form of a function to display its key features.

That last testing point is vertex form in all but name.

Three forms, three different things they show

In every form, the sign of a tells you which way the parabola opens: upwards with a minimum when a > 0, downwards with a maximum when a < 0.

How to rewrite a quadratic in vertex form

Completing the square is the reliable method. Take f(x) = x² − 6x + 5. Half of −6 is −3, and (−3)² = 9, so add and subtract 9:

f(x) = (x² − 6x + 9) − 9 + 5 = (x − 3)² − 4

The vertex is (3, −4), and because a = 1 is positive, the minimum value of f is −4, reached at x = 3. As a check, the factored form is (x − 1)(x − 5): the zeros are 1 and 5, and 3 lies exactly halfway between them.

When a is not 1, factor it out of the x terms first. For g(x) = 2x² + 8x + 3:

g(x) = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5

The vertex is (−2, −5). Notice the sign: (x + 2)² means h = −2, not 2.

If you only need the vertex, there is a shortcut: its x-coordinate is −b/(2a). For g, that is −8/4 = −2, and g(−2) = 8 − 16 + 3 = −5, the same answer in two lines.

The question types you will meet

1. Find the minimum or maximum

For h(x) = −(x − 4)² + 9, a = −1 is negative, so the parabola opens downwards and the maximum value is 9, at x = 4. The classic wrong answer is 4: the x-value where the maximum occurs, not the maximum itself. College Board's own sample explanations point out exactly this distractor.

2. Choose the form that displays a feature

Suppose a question gives f(x) = (x − 2)(x − 8) and asks which equivalent form shows the minimum value of f as a constant or coefficient. The zeros are 2 and 8, so the vertex is at x = 5, and f(5) = (3)(−3) = −9. The answer is (x − 5)² − 9. If the question had asked for a form showing the x-intercepts, the factored form would already be the answer.

3. Count the real solutions

The discriminant b² − 4ac decides how many real solutions a quadratic equation has: two if it is positive, one if it is zero, none if it is negative. Suppose x² − 6x + c = 0 has exactly one real solution. Then 36 − 4c = 0, so c = 9. Check: x² − 6x + 9 = (x − 3)², which is zero only at x = 3. If the question asked for no real solutions instead, the answer would be any c greater than 9.

4. Where a line meets a parabola

Take the system y = x² − 2x − 3 and y = x + 1. Set the right-hand sides equal: x² − 3x − 4 = 0, which factorises as (x − 4)(x + 1) = 0. The graphs meet at (4, 5) and (−1, 0). A question may ask for the points, for one coordinate, or only for how many intersections there are, which is the discriminant question again.

5. Quadratics in context

A ball's height in metres, t seconds after it is thrown, is h(t) = −5t² + 20t + 1. The vertex is at t = −20/(2 × −5) = 2, and h(2) = −20 + 40 + 1 = 21, so the ball reaches a maximum height of 21 metres after 2 seconds. In vertex form, h(t) = −5(t − 2)² + 21. Context questions can ask what a number means rather than for a calculation: here the constant 1 is the height at t = 0, the moment the ball is thrown.

6. Shifted graphs

The graph of y = (x + 1)² − 2 is the graph of y = x² moved 1 unit left and 2 units down, so its vertex is (−1, −2). College Board's testing points for shifted graphs are about connecting an equation with a graph or a table, and vertex form makes that connection immediate.

7. Typing in your own answer

About a quarter of Math questions are student-produced responses, where you enter the answer yourself, and College Board notes that some of them have more than one correct answer even though you enter only one. If you are asked for a solution of x² − 5x + 6 = 0, both 2 and 3 are correct, since the equation factorises as (x − 2)(x − 3) = 0.

Using the built-in calculator

A Desmos graphing calculator is built into Bluebook for the whole Math section. Graphing a quadratic shows its vertex and intercepts at once, and graphing two equations shows where they meet. Use it to check algebra or to answer a question that only needs a value. It will not, however, tell you which form a question wants or what a constant means in context, and those are the questions where careful reading matters most.

Mistakes that cost points

• The sign of h. In (x + 7)², the vertex is at x = −7.
• Completing the square with a left inside. Factor a out of the x terms first, then multiply it back in at the end.
• Answering with the wrong coordinate. The minimum value is k; the place where it happens is h.
• Forgetting a in the discriminant. For 3x² + 6x + c, it is 36 − 12c, not 36 − 4c.

How to practise quadratics

Work skill by skill rather than topic by topic. College Board's Student Question Bank can be filtered by domain, skill and difficulty, and so can our SAT question bank, which follows College Board's domains: its Advanced Math skills include Equivalent expressions, Nonlinear equations and systems, and Nonlinear functions, each with easy, medium and hard questions and a worked explanation for every answer. Start at the level you can mostly solve, then move up once the methods above feel automatic.

Then check the skill under pressure. Advanced Math questions appear in both Math modules, mixed with the other domains, so the real test of whether vertex form has stuck is a timed, full-length sitting. Our full-length SAT practice tests report accuracy by topic, so you can see whether Advanced Math has moved, and our Digital SAT practice page explains how the practice mirrors the real test.

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