A quadratic function is a polynomial of degree 2, written f(x) = ax² + bx + c, whose average rates of change over equal-length intervals form a linear pattern. In AP Precalculus it models data that is roughly symmetric with a unique maximum or minimum, like projectile height or area.
A quadratic function is a polynomial function of degree 2, in the form f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola with a single turning point, which is either an absolute maximum (if a < 0) or an absolute minimum (if a > 0).
Here's the fingerprint AP Precalculus actually cares about. Take a quadratic and compute its average rate of change over consecutive equal-length intervals. Those rates won't be constant (that would be linear), but they change by the same amount each time. In CED language, the average rates of change of a quadratic can be given by a linear function (EK 1.3.A.2), so they're changing at a constant rate (EK 1.3.B.2). Constant second differences, linear first differences. That pattern is how you identify a quadratic from a table without ever seeing the equation.
Quadratic functions anchor three topics in Unit 1. In Topic 1.3, learning objectives 1.3.A and 1.3.B use quadratics as the first step up from linear functions in the rate-of-change hierarchy. Linear functions have constant rates of change; quadratics have linearly changing rates of change. That ladder keeps extending all the way to calculus, so internalizing it now pays off later.
In Topics 1.13 and 1.14 (LOs 1.13.A, 1.14.A, 1.14.B), quadratics are one of your core modeling tools. The CED tells you exactly when to reach for one. Pick a quadratic when the data shows roughly linear rates of change, when the data is roughly symmetric with a unique max or min, or when the context involves area or two dimensions (EK 1.13.A.2 and 1.13.A.3). Model selection questions are essentially asking you to recognize these clues.
Keep studying AP® Precalculus Unit 1
nth differences (Unit 1)
A quadratic has constant second differences over equal-length input intervals. This is the table-based test for degree. If first differences are constant it's linear, if second differences are constant it's quadratic, and the pattern generalizes to any polynomial degree.
Quadratic regression (Unit 1)
When real data is only roughly parabolic, you don't solve for a, b, and c by hand. You run a quadratic regression on your calculator to find the best-fit quadratic model, then use it to predict values (LO 1.14.B).
Cubic function (Unit 1)
The CED draws a clean dimensional line. Area and two-dimensional contexts suggest quadratic models, while volume and three-dimensional contexts suggest cubic models (EK 1.13.A.3). Cubics also have linearly changing second differences, one step up the ladder.
Absolute maximum (Unit 1)
A quadratic's vertex is a guaranteed absolute max or min, which makes quadratics the go-to model for optimization contexts like maximum height, maximum revenue, or maximum area.
Multiple-choice questions love the rate-of-change fingerprint. A classic stem gives you average rates of change like -7, -3, 1, 5 over consecutive equal intervals and asks you to extend the pattern (each rate jumps by 4, so the next is 9). Harder versions go in reverse and hand you the rates so you can reconstruct the actual quadratic g(x), or give you the rates as a linear function L(t) = 5t - 2 and ask for the constant rate at which the rates increase (it's the slope, 5).
On the free-response side, modeling FRQs like 2025 FRQ Q2 (a song's streams over time) give you real-world data and ask you to build a function model, justify why that function type fits, and use it to answer questions in context with correct units. For quadratics, your justification language is straight from EK 1.13.A.2. Say the data has roughly linear rates of change, or that it's roughly symmetric with a unique maximum or minimum. Also be ready to state domain restrictions, since negative time or negative streams usually make no sense in context (EK 1.13.B.3).
The phrase "linear rates of change" trips everyone up. A linear function has a CONSTANT rate of change. A quadratic function has rates of change that are themselves LINEAR, meaning they increase or decrease by the same amount each interval. If an MCQ says the average rates of change follow a linear pattern, the function is quadratic, not linear. One level of "changing" bumps the degree up by one.
A quadratic function is a degree-2 polynomial f(x) = ax² + bx + c whose graph is a parabola with one absolute maximum or minimum.
Over consecutive equal-length intervals, a quadratic's average rates of change form a linear pattern, so those rates change at a constant rate (EK 1.3.A.2 and 1.3.B.2).
In a table with equal-length inputs, constant second differences mean the function is quadratic.
Choose a quadratic model when data is roughly symmetric with a unique max or min, when rates of change are roughly linear, or when the context involves area or two dimensions (EK 1.13.A.2, 1.13.A.3).
On modeling FRQs, you may build the quadratic by hand, by transforming the parent function, or with quadratic regression, then answer context questions with appropriate units and domain restrictions.
It's a polynomial function of degree 2, written f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola, and AP Precalc identifies it by its average rates of change, which change at a constant rate over equal-length intervals.
No. Linear functions have a constant rate of change. A quadratic's average rates of change over consecutive equal-length intervals follow a linear pattern, so the rates themselves change at a constant rate. That distinction is exactly what Topic 1.3 tests.
Check the differences over equal-length inputs. If first differences are constant, it's linear. If first differences change by the same amount each time (constant second differences), it's quadratic. For example, rates of -3, 1, 5, 9 each jump by 4, so the function is quadratic.
The CED gives a dimensional rule of thumb. Area and two-dimensional contexts usually call for quadratic models, while volume and three-dimensional contexts usually call for cubic models (EK 1.13.A.3). Also pick quadratic when data is roughly symmetric with one max or min.
Solving ax² + bx + c = 0 is assumed background skill, but the exam's focus is different. You're tested on recognizing quadratics from rate-of-change patterns, justifying quadratic models from data, and using those models to predict values and rates in context.
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