SAT Quadratic Functions: Factoring, Vertex Form & Graphs

Quadratic functions are among the most tested Advanced Math topics on the Digital SAT — often about 4–6 items across modules. You will solve for roots, read or build vertex form, connect equations to graphs, and use the discriminant to count solutions.

The test loves switching forms: standard ax² + bx + c, factored a(x − r)(x − s), and vertex a(x − h)² + k. The same parabola can be “easy” in one form and slow in another. Elite pacing is knowing which form answers which question fastest.

Context items also appear: projectile height, profit, area models. The algebra is quadratic; the trap is answering the wrong feature (time of max height vs. max height value).

Standard form y = ax² + bx + c: good for quadratic formula, discriminant, and y-intercept (c). Opening up if a > 0, down if a < 0.

Factored form y = a(x − r)(x − s): roots/x-intercepts at x = r and x = s are immediate. Axis of symmetry is midway between roots.

Vertex form y = a(x − h)² + k: vertex at (h, k) is immediate. Max/min value is k; axis of symmetry is x = h. Remember (x − 3)² means h = +3, not −3.

Before computing, ask: “Do I need roots, vertex/extremum, or number of solutions?” Pick the form (or conversion) that makes that feature free.

Solve: x² − 5x + 6 = 0.

Find two numbers that multiply to 6 and add to −5: −2 and −3. Factor: (x − 2)(x − 3) = 0. Roots: x = 2 or x = 3.

Harder: 2x² − 7x − 4 = 0. Look for factors of (2)(−4) = −8 that add to −7: −8 and +1. Split the middle: 2x² − 8x + x − 4 = 0 → 2x(x − 4) + 1(x − 4) = 0 → (2x + 1)(x − 4) = 0. Roots: x = −1/2 or x = 4.

Try factoring first on every clean-looking quadratic. Formula is backup when integers do not cooperate.

Solve: x² + 4x + 1 = 0. a = 1, b = 4, c = 1.

Discriminant D = b² − 4ac = 16 − 4 = 12 > 0 → two real roots.

x = (−4 ± √12) / 2 = (−4 ± 2√3) / 2 = −2 ± √3.

Sign discipline: the formula starts with −b. If b = −5, then −b = +5. Write −b as its own step.

Discriminant-only items: D > 0 two real, D = 0 one real (double root / vertex on x-axis), D < 0 no real roots. You may never need the full ± expression.

A ball’s height in meters is h(t) = −5(t − 2)² + 45, with t in seconds. What is the maximum height, and when does it occur?

Already vertex form: a = −5 < 0 so the parabola opens down and the vertex is a maximum. Vertex at (h, k) = (2, 45). Maximum height is 45 m at t = 2 s.

Standard-form path: for y = ax² + bx + c, vertex x-coordinate is x = −b/(2a), then plug in for y. Completing the square converts standard to vertex when you need (h, k) without formula memorization: x² + 6x + 5 = (x + 3)² − 4, vertex (−3, −4).

Trap: answering t = 2 when the question asked for the height, or answering 45 when it asked for the time. Box the asked quantity.

Suppose a graph shows a parabola with x-intercepts at −1 and 3 and a y-intercept at −3. Which factored form could match?

Roots at −1 and 3 → factors (x + 1)(x − 3). So y = a(x + 1)(x − 3). Use the y-intercept: when x = 0, y = a(1)(−3) = −3a. Set −3a = −3 → a = 1. Equation: y = (x + 1)(x − 3).

Axis of symmetry would be midway: x = (−1 + 3)/2 = 1. Vertex lies on that line; you can find y by substituting x = 1 if needed.

When a graph question only asks for the axis or the vertex x-value, do not solve a full system of coefficients unless required.

Order of attack for solving: (1) factor if obvious, (2) formula if not, (3) complete the square when vertex form is the real goal.

For graph ↔ equation items, list known features (roots, vertex, direction) before looking at choices. Eliminate forms that contradict one feature.

Use symmetry: axis is midway between roots; vertex sits on the axis.

Calculator/Desmos: graphing can confirm roots or vertex, but for clean factorable items algebra is faster. Use graphs when coefficients are ugly or the question is already visual.

Always re-read whether the ask is a root, a vertex coordinate, a max value, or a parameter constraint.

Vertex sign errors: (x + 4)² has h = −4. Say out loud “what x makes the square zero?”

Quadratic formula −b mistakes when b is negative.

Dropping the ± and reporting only one root when two are required.

Confusing “no real solutions” with “no solutions in the answer choices” — still use the discriminant.

Context mix-ups: reporting the time when asked for height (or the reverse).

Expanding carefully: a(x − h)² + k mistakes when distributing a only to the square and forgetting a multiplies the expanded terms correctly — expand step by step on paper.

Day 1–2: Factoring and roots only, including a ≠ 1.

Day 3: Quadratic formula + discriminant-only items.

Day 4: Vertex form reading and completing the square to vertex form.

Day 5: Graph feature matching and “which equation matches the graph.”

Day 6: Mixed 12–15; tag content vs careless (especially sign and asked-quantity errors).

Day 7: Timed mini-set, then convert every miss into the form you should have used.

Ready bar: convert among three forms without notes; state vertex from vertex form in one glance; choose factor vs formula in under five seconds.

Advanced Math topic sessions mix forms so you practice choosing a method under mixed order — closer to Module 2 than a textbook chapter that only factors for ten pages.

On misses, ask the AI tutor (Pro) “which form makes this feature free?” or “show only the completing-the-square steps.” Narrow prompts fix the exact leak.

Connect to Polynomials & Factoring for factor patterns, and to Exponential Functions when growth models start appearing beside quadratics in Advanced Math sets.