A quadratic function is a polynomial function of degree 2 written in standard form as f(x) = ax² + bx + c (where a ≠ 0). Its graph is a parabola — a U-shaped curve that opens upward when a > 0 and downward when a < 0. Quadratic functions have three equivalent forms (standard, vertex, and factored), are solved using factoring, completing the square, or the quadratic formula, and appear throughout Florida MAFS Algebra standards and the SAT Math “Advanced Math” section.
Function that looks like a “u” when graphed. Has 3 forms standard, factor and vertex form.
f(x) = ax² + bx + c
Use to: find y-intercept (c) · evaluate discriminant · apply quadratic formula
f(x) = a(x - h)² + k
Use to: identify vertex (h, k) · find axis of symmetry x = h · graph transformations
f(x) = a(x - r₁)(x - r₂)
Use to: find x-intercepts r₁ and r₂ directly · determine roots/zeros quickly
a ≠ 0 · a determines direction (a > 0: opens up; a < 0: opens down) · c = y-intercept (the constant term) · Vertex x-coordinate: x = -b/(2a) · Use this form for: quadratic formula, discriminant analysis, finding y-intercept. Most common form on Florida EOC assessments.
Vertex is at (h, k) – the minimum (a > 0) or maximum (a < 0) point of the parabola · Axis of symmetry: x = h · SAT key insight: when the problem asks for the maximum or minimum value of a quadratic function, convert to or recognize vertex form – the answer is k. Most common form on SAT "Advanced Math" parabola questions.
r₁ and r₂ are the roots (zeros, x-intercepts, solutions) – set each factor = 0 to find them · The sum of roots: r₁ + r₂ = -b/a · The product of roots: r₁ × r₂ = c/a (Vieta's formulas – a high-difficulty SAT concept) · Only works when the quadratic is factorable over integers. If discriminant b²–4ac is not a perfect square, use the quadratic formula instead.
Works for ANY quadratic equation ax²+bx+c = 0 · The expression b²–4ac under the radical is called the discriminant:
b²–4ac > 0 → two distinct real roots (parabola crosses x-axis twice)
b²–4ac = 0 → exactly one real root (parabola touches x-axis at vertex)
b²–4ac < 0 → no real roots (parabola does not cross x-axis) · On SAT: this formula is NOT provided on the reference sheet – memorize it.
The turning point of the parabola — the maximum if a 0. From vertex form: read (h, k) directly. From standard form: x = −b/(2a) gives h; substitute back to get k = f(−b/2a). The vertex is the most frequently tested parabola feature on SAT Advanced Math.
The vertical line x = h that divides the parabola symmetrically into two mirror halves. A parabola is perfectly symmetric about this vertical line. From standard form: x = −b/(2a). From vertex form: x = h. On the SAT: if you know one root at x = 3 and the axis is x = 5, the other root must be x = 7 (equidistant from x = 5).
Where the parabola crosses the y-axis — found by setting x = 0. In standard form f(x) = ax²+bx+c: the y-intercept is always c (the constant term). This is the simplest feature to read from standard form and a frequent 10-second SAT question.
Where the parabola crosses the x-axis — also called roots, zeros, or solutions. Found by factoring (factored form), or the quadratic formula (standard form). The number of x-intercepts is determined by the discriminant: b²−4ac > 0 → two; = 0 → one (tangent); < 0 → zero. On SAT: "How many real solutions?" always means: evaluate the discriminant.
| DISCRIMINANT VALUE | NUMBER OF REAL ROOTS | GRAPH BEHAVIOR | SAT IMPLICATION |
|---|---|---|---|
| b²–4ac > 0 | Two distinct real roots | Parabola crosses x-axis twice | Two x-intercepts visible on graph; factoring or QF gives two answers |
| b²–4ac = 0 | Exactly one real root | Parabola touches x-axis at vertex | Vertex is ON the x-axis; root = -b/(2a); SAT: "tangent to the x-axis" |
| b²–4ac < 0 | No real roots | Parabola does not cross x-axis | No x-intercepts; "How many real solutions?" answer = 0; complex roots exist |
| SAT QUESTION TYPE | WHAT IT TESTS | FREQUENCY |
|---|---|---|
| Vertex / Maximum / Minimum | Identify vertex (h,k) from vertex form or convert from standard form; determine max or min value and where it occurs | 3–4 per test |
| Number of Solutions | Use discriminant b²–4ac to determine how many real solutions a quadratic equation has; "How many x-intercepts?" | 2–3 per test |
| Roots / Zeros | Solve by factoring or quadratic formula; identify x-intercepts from factored form; use Vieta's formulas (sum/product of roots) | 2–3 per test |
| Form Conversion | Convert between standard, vertex, and factored form; complete the square; identify equivalent expressions | 1–2 per test |
| System: Linear + Quadratic | Find intersection points of a line and a parabola; substitute linear into quadratic; use discriminant to find tangency condition | 1 per test |
The quadratic formula is x = (−b ± √(b²−4ac)) / 2a. It is used to find the roots (solutions) of any quadratic equation in standard form ax²+bx+c = 0. Use it when the quadratic cannot be factored easily — specifically when the discriminant b²−4ac is not a perfect square. The quadratic formula always works, regardless of whether the quadratic is factorable.
Standard form (ax²+bx+c) is best for finding the y-intercept (the constant c) and applying the quadratic formula or discriminant. Vertex form (a(x−h)²+k) directly reveals the vertex (h, k) and is best for finding maximum or minimum values and graphing transformations. Factored form (a(x−r₁)(x−r₂)) directly reveals the x-intercepts (r₁ and r₂). On the SAT, knowing which form reveals which feature instantly saves 60–90 seconds per question.
The discriminant is the expression b²−4ac found under the radical in the quadratic formula. It tells you the number of real solutions: if b²−4ac > 0, the quadratic has two distinct real roots (the parabola crosses the x-axis twice); if b²−4ac = 0, there is exactly one real root (the vertex touches the x-axis); if b²−4ac < 0, there are no real roots (the parabola does not cross the x-axis and the solutions are complex numbers).
Yes — extensively. Quadratic functions are the most frequently tested topic in the SAT Math “Advanced Math” category, which accounts for approximately 35% of all SAT Math questions. The SAT does NOT provide the quadratic formula on its reference sheet. Common SAT question types include: finding the vertex or maximum/minimum, using the discriminant to count solutions, solving systems of a line and a parabola, and converting between quadratic forms. Florida students preparing for Bright Futures scholarship score requirements must be proficient with all of these question types.
Quadratic functions are covered in multiple Florida MAFS standards: MAFS.912.A-APR.B.3 (identify zeros of polynomials and use them to construct a rough graph of the function), MAFS.912.A-REI.B.4 (solve quadratic equations in one variable by taking square roots, completing the square, the quadratic formula, and factoring), and MAFS.912.F-IF.C.8 (write a function in different but equivalent forms to reveal and explain different properties). These standards are assessed on the Florida EOC Algebra 1 and Algebra 2 exams taken by students in Orlando, Winter Park, Lake Nona, and surrounding communities.