Overview
Quadratic functions produce parabola-shaped graphs and appear extensively in SAT Advanced Math, both as standalone function questions and in real-world modeling contexts. The SAT presents quadratic functions in three interchangeable forms, and the ability to read information directly from each form — and convert between forms efficiently — is a high-yield skill. The digital SAT’s Desmos calculator makes graphical verification fast and reliable.
Key Points
1. Three Forms of a Quadratic Function
| Form | Equation | What It Reveals Directly |
|---|---|---|
| Standard | f(x) = ax² + bx + c | y-intercept (c), direction (sign of a) |
| Vertex | f(x) = a(x - h)² + k | Vertex (h, k), axis of symmetry x = h |
| Factored | f(x) = a(x - p)(x - q) | x-intercepts (roots) p and q |
The value of a is shared across all three forms and determines:
- a > 0 → parabola opens upward (minimum exists)
- a < 0 → parabola opens downward (maximum exists)
- |a| larger → narrower parabola; |a| smaller → wider parabola
2. Vertex and Axis of Symmetry
From standard form:
Axis of symmetry: x = -b / (2a)
Vertex x-coordinate: h = -b / (2a)
Vertex y-coordinate: k = f(h) = substitute h back into f(x)
From vertex form:
f(x) = a(x - h)² + k
Vertex is directly (h, k)
Axis of symmetry is x = h
Note: In vertex form, the sign inside the parentheses is opposite to h. If the form is f(x) = a(x + 3)² + k, then h = -3 (not +3).
3. Converting Between Forms
Standard → Vertex (complete the square):
f(x) = x² + 6x + 5
= (x² + 6x + 9) - 9 + 5
= (x + 3)² - 4
Vertex: (-3, -4)
Factored → Standard (expand):
f(x) = 2(x - 1)(x - 4) = 2(x² - 5x + 4) = 2x² - 10x + 8
Standard → Factored (factor):
f(x) = x² - 5x + 6 = (x - 2)(x - 3) → roots: x = 2, 3
4. Key Graph Features
| Feature | How to Find |
|---|---|
| y-intercept | Set x = 0; equals c in standard form |
| x-intercepts (roots) | Set f(x) = 0; use factoring or quadratic formula |
| Vertex | x = -b/2a; or read directly from vertex form |
| Axis of symmetry | x = h (same as vertex x-coordinate) |
| Maximum/Minimum value | y-coordinate of vertex; min if a>0, max if a<0 |
5. Maximum and Minimum in Context
On the SAT, word problems about quadratic functions often ask for:
- Maximum profit → find vertex y-coordinate when a < 0
- Maximum height → find vertex y-coordinate of a projectile equation
- When max/min occurs → find vertex x-coordinate
Example pattern:
“The height h (in feet) of a ball is given by h(t) = -16t² + 64t + 5. What is the maximum height?“
t_vertex = -64 / (2·(-16)) = -64 / -32 = 2 seconds
h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = 69 feet
6. Digital SAT: Desmos Strategy
- Graph the quadratic → identify vertex, x-intercepts, direction directly
- Regression method (3 points given): open Desmos table, enter (x, y) pairs, type
y₁ ~ ax₁² + bx₁ + cto fit - Solve for intersections: graph quadratic and any other equation, click intersection points
Pitfalls and Common Mistakes
Pitfall 1: Sign error reading h from vertex form f(x) = a(x + 3)² + k → students read h = +3, but correct h = -3. Fix: Rewrite as f(x) = a(x - (-3))² + k. The vertex is (-3, k), not (3, k).
Pitfall 2: Confusing axis of symmetry with a solution The axis of symmetry x = -b/(2a) is NOT a solution to the equation — it is the x-coordinate of the vertex. Fix: Clearly distinguish between “vertex x-coordinate” and “roots of the equation.”
Pitfall 3: Forgetting the negative sign for downward parabolas If a < 0, the vertex is a maximum, not a minimum. Fix: Always note the sign of a first before labeling vertex as max or min.
Pitfall 4: Wrong y-intercept In vertex form f(x) = a(x-h)² + k, students write the y-intercept as k. Fix: Substitute x = 0: y-intercept = a(0-h)² + k = ah² + k.
Pitfall 5: Expanding vertex form incorrectly
WRONG: a(x - h)² = ax² - ah²
RIGHT: a(x - h)² = a(x² - 2hx + h²) = ax² - 2ahx + ah²
Fix: Always expand (x - h)² = x² - 2hx + h² fully before multiplying by a.
Related Entries
- Quadratic_Equations
- Function_Transformations
- Systems_Nonlinear_Equations
- Functions_Concepts_Notation
- Polynomial_Operations_Factoring
Quick Reference Card
| Concept | Formula / Rule |
|---|---|
| Standard form | f(x) = ax² + bx + c |
| Vertex form | f(x) = a(x - h)² + k |
| Factored form | f(x) = a(x - p)(x - q) |
| Axis of symmetry | x = -b/(2a) |
| Vertex from standard | (-b/2a, f(-b/2a)) |
| Opens up | a > 0 (minimum at vertex) |
| Opens down | a < 0 (maximum at vertex) |
| y-intercept | (0, c) in standard form |
| x-intercepts | Roots of ax²+bx+c=0 |
| Vertex h from vertex form | h is opposite sign: f(x-h) means h is positive |