AP-PRECALC
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Expected availability: Summer 2026
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AP® Precalculus
AP Precalculus teaches polynomial, rational, exponential, logarithmic, trigonometric, polar, and parameter‑based functions, emphasizing graphing, algebraic manipulation, and theorem application to prepare students for calculus and analytical reasoning.
180
Minutes
44
Questions
3/5
Passing Score
$98
Exam Cost
Who Should Take This
High‑school juniors and seniors who have completed Algebra II and are planning to enroll in AP Calculus should take this exam. It suits motivated learners seeking a rigorous foundation in function theory, vector and matrix concepts, and the ability to translate mathematical ideas across graphical, numerical, and verbal representations.
What's Covered
1
All four units of the AP Precalculus course framework (College Board, effective 2023-present): Unit 1 Polynomial and Rational Functions
2
, Unit 2 Exponential and Logarithmic Functions
3
, Unit 3 Trigonometric and Polar Functions
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
20 Activity Formats
Course Outline
61 learning goals
1
Unit 1: Polynomial and Rational Functions
4 topics
Rates of Change and Polynomial Behavior
- Define average rate of change of a function over an interval and calculate it from a table, graph, or algebraic expression.
- Explain how the rate of change of a function varies across different intervals and relate increasing or decreasing rates of change to concavity of the function's graph.
- Analyze second differences in a table of values to determine whether a data set is best modeled by a linear, quadratic, or higher-degree polynomial function.
- Explain what it means for a function to be increasing, decreasing, or constant on an interval and identify these intervals from the graph or algebraic expression of a function.
Polynomial Functions
- Identify the degree, leading coefficient, and constant term of a polynomial function and state the relationship between degree and maximum number of real zeros and turning points.
- Determine the real and complex zeros of polynomial functions using factoring, the rational root theorem, and synthetic division, and express the polynomial in factored form.
- Describe the end behavior of polynomial functions based on the sign and parity of the leading term and sketch approximate graphs using zeros and end behavior.
- Explain the multiplicity of a zero and describe how even and odd multiplicities affect whether the graph crosses or touches the x-axis at that zero.
- Construct a polynomial function of specified degree given its zeros (including multiplicity) and a point on the graph to determine the leading coefficient.
- Apply the remainder theorem and factor theorem to evaluate polynomials at specific inputs and determine whether a given value is a zero of the polynomial.
Rational Functions
- Define a rational function as a ratio of two polynomials and identify its domain restrictions where the denominator equals zero.
- Determine the vertical asymptotes and holes of a rational function by factoring numerator and denominator and identifying common versus non-common zero factors.
- Determine the horizontal or slant asymptote of a rational function by comparing the degrees of the numerator and denominator and performing polynomial long division when appropriate.
- Sketch the graph of a rational function showing all intercepts, asymptotes, and holes, and describe the function's behavior near each feature.
- Solve rational equations algebraically and verify solutions by checking for extraneous roots introduced by multiplying both sides by variable expressions.
Function Transformations and Composition
- Describe the effects of vertical and horizontal translations, reflections, and stretches on the graph of a function using transformation notation y = a*f(b(x-h)) + k.
- Determine the domain and range of a transformed function given the domain and range of the original function and the applied transformations.
- Compose two functions (f composed with g) algebraically and determine the domain of the composition by considering the domains of both component functions.
- Determine whether a function has an inverse by applying the horizontal line test, and find the inverse function algebraically by solving for the input variable.
- Identify even and odd functions using algebraic tests (f(-x) = f(x) for even, f(-x) = -f(x) for odd) and describe the graphical symmetry each property produces.
- Analyze piecewise-defined functions by evaluating each piece on its specified domain, graphing the complete function, and identifying any discontinuities.
2
Unit 2: Exponential and Logarithmic Functions
3 topics
Exponential Functions
- Define exponential functions f(x) = a*b^x and identify their key properties including domain, range, asymptote, and whether the function models growth or decay based on the base.
- Explain how constant percentage change per unit interval characterizes exponential functions and distinguish this behavior from the constant additive change of linear functions.
- Construct exponential models from real-world data involving population growth, radioactive decay, compound interest, or temperature change using given initial values and growth/decay rates.
- Apply transformations to exponential functions (shifts, reflections, stretches) and describe how each transformation changes the graph, asymptote, and range.
- Explain continuous compounding using the formula A = Pe^(rt) and compare the results of continuous versus discrete compounding at various frequencies.
Logarithmic Functions
- Define logarithmic functions as inverses of exponential functions and convert fluently between exponential form (b^y = x) and logarithmic form (log_b(x) = y).
- Apply the properties of logarithms (product, quotient, power rules) to expand, condense, and simplify logarithmic expressions.
- Solve exponential equations by applying logarithms to both sides and solve logarithmic equations by converting to exponential form, checking for extraneous solutions.
- Describe the graph of a logarithmic function including its domain, range, vertical asymptote, and the effect of base value on the rate of growth.
- Apply the change-of-base formula to evaluate logarithms with any base using natural or common logarithms and verify the result numerically.
- Construct a logarithmic model from real-world data (earthquake magnitude, sound intensity, pH) and interpret the meaning of the logarithmic scale in context.
Semi-Log Plots and Data Linearization
- Explain how plotting data on a semi-log scale linearizes exponential relationships and how the slope and intercept of the resulting line relate to the exponential model parameters.
- Analyze a data set by creating a semi-log plot to determine whether an exponential model is appropriate and extract the model parameters from the linearized data.
- Compare the residual plots of linear, quadratic, and exponential models fitted to the same data set and select the model that best represents the underlying relationship.
3
Unit 3: Trigonometric and Polar Functions
4 topics
Trigonometric Functions on the Unit Circle
- Define the six trigonometric functions using the unit circle and state the exact values of sine, cosine, and tangent at standard angles (0, pi/6, pi/4, pi/3, pi/2 and their multiples).
- Convert between degree and radian measure and explain why radians are the natural unit of angle measurement for trigonometric functions.
- Explain the periodicity of trigonometric functions and determine the sign of each function in each quadrant using the ASTC (All Students Take Calculus) framework.
Graphs of Trigonometric Functions
- Identify the amplitude, period, phase shift, and vertical shift of sinusoidal functions in the form y = A sin(B(x - C)) + D and describe their effect on the graph.
- Sketch the graphs of sine, cosine, and tangent functions including transformed versions, labeling key features such as maxima, minima, zeros, and asymptotes.
- Construct a sinusoidal model from real-world periodic data (tides, temperature, daylight hours) by determining amplitude, period, and vertical shift from the data.
- Compare the graphs of reciprocal trigonometric functions (secant, cosecant, cotangent) to their parent functions and identify their asymptotes and domains.
Inverse Trigonometric Functions and Identities
- Define the inverse trigonometric functions (arcsin, arccos, arctan) by specifying the restricted domains that make the original functions one-to-one.
- Evaluate compositions of trigonometric and inverse trigonometric functions, including expressions like sin(arccos(x)), by constructing reference triangles.
- Verify and apply fundamental trigonometric identities including Pythagorean identities (sin^2 + cos^2 = 1) and sum/difference formulas for sine and cosine.
- Solve trigonometric equations over specified intervals by applying identities, algebraic techniques, and inverse functions to find all solutions.
- Apply the double-angle formulas for sine and cosine (sin 2x = 2 sin x cos x, cos 2x = cos^2 x - sin^2 x) to simplify expressions and solve equations.
- Apply the law of sines and law of cosines to solve oblique triangles and determine the number of possible solutions in the ambiguous case.
Polar Coordinates and Functions
- Convert between polar coordinates (r, theta) and Cartesian coordinates (x, y) using the relationships x = r cos(theta) and y = r sin(theta).
- Identify and sketch the graphs of standard polar curves including circles, cardioids, rose curves, and limacons, describing how the parameters affect the shape.
- Analyze the rates of change of polar functions r(theta) and explain how the rate at which r changes with theta determines the shape and symmetry of the polar curve.
4
Unit 4: Functions Involving Parameters, Vectors, and Matrices
3 topics
Parametric Functions
- Define parametric equations as functions x(t) and y(t) that define a curve in the plane and plot parametric curves by evaluating at multiple values of the parameter.
- Eliminate the parameter from a pair of parametric equations to obtain a Cartesian equation and identify the resulting curve type (line, parabola, circle, ellipse).
- Construct parametric equations to model projectile motion and other real-world scenarios involving independent horizontal and vertical components of movement.
Vectors
- Define vectors in two dimensions, represent them in component form and magnitude-direction form, and perform vector addition, subtraction, and scalar multiplication.
- Calculate the magnitude of a vector and the dot product of two vectors, and use the dot product to determine the angle between vectors.
- Apply vectors to solve problems involving displacement, velocity, and force by decomposing vectors into components and combining them using vector addition.
Matrices and Transformations
- Define matrices and perform matrix addition, scalar multiplication, and matrix multiplication for 2x2 and 2x1 matrices, verifying that multiplication is not commutative.
- Represent geometric transformations (rotation, reflection, dilation) as 2x2 matrix multiplications and apply them to transform points and shapes in the coordinate plane.
- Compose multiple geometric transformations by multiplying their corresponding matrices and interpret the resulting transformation geometrically.
- Calculate the inverse of a 2x2 matrix and explain how inverse matrices reverse a transformation, connecting this to the concept of function inverses.
Scope
Included Topics
- All four units of the AP Precalculus course framework (College Board, effective 2023-present): Unit 1 Polynomial and Rational Functions (30-40%), Unit 2 Exponential and Logarithmic Functions (27-40%), Unit 3 Trigonometric and Polar Functions (15-20%), Unit 4 Functions Involving Parameters, Vectors, and Matrices (optional, not assessed on AP exam).
- Polynomial and rational functions: rates of change, polynomial behavior and zeros, end behavior, transformations, rational functions including asymptotes and holes, and modeling with polynomial and rational functions.
- Exponential and logarithmic functions: exponential growth and decay models, logarithmic functions and their properties, transformations, solving exponential and logarithmic equations, and semi-log plots for data analysis.
- Trigonometric and polar functions: trigonometric ratios on the unit circle, graphs of sine, cosine, and tangent functions, sinusoidal models, inverse trigonometric functions, trigonometric identities, and polar coordinates with polar function graphs.
- Functions involving parameters, vectors, and matrices: parametric functions, vectors in two dimensions, and matrices as transformations (exploratory unit, not on AP exam).
- Function analysis throughout all units: domain and range, function composition, inverses, transformations (vertical/horizontal shifts, stretches, reflections), piecewise functions, and function modeling.
- Data analysis and mathematical modeling: using functions to model real-world data, constructing and interpreting residual plots, and selecting appropriate function types based on context.
Not Covered
- Limits, derivatives, integrals, and other calculus concepts covered in AP Calculus AB/BC.
- Advanced linear algebra, eigenvalues, and matrix theory beyond basic 2x2 transformations in Unit 4.
- Complex analysis, Fourier series, and advanced trigonometric analysis beyond the AP Precalculus framework.
- Probability and statistics topics covered in AP Statistics.
Official Exam Page
Learn more at College Board
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