The rate-of-change, polynomial, exponential, logarithmic, trigonometric, and polar facts worth having memorized for AP Precalculus, organized by unit and annotated with what each one is for. These are the standard, well-known results, not an official College Board formula sheet; confirm the exam's exact provided materials and calculator policy on AP Central. Every entry links to the unit where Mistake Master teaches and drills it.
Unit 1
Polynomial and Rational Functions
How functions change and how their structure shows on a graph.
$$ \text{AROC} = \dfrac{f(b) - f(a)}{b - a} $$
Average rate of change
The slope of the secant line across an interval. It is not the rate at either endpoint.
Unit 1 · Topic 1.2
$$ \text{concave up} \iff \text{rate of change is increasing} $$
Concavity
Concave up means the rate of change is increasing; concave down means it is decreasing. This is separate from whether the function itself is increasing.
Unit 1 · Topic 1.1
$$ (x - a)^m $$
Zero of multiplicity m
At a real zero x = a of multiplicity m, the graph crosses the x-axis when m is odd and touches then turns when m is even.
The end behavior of a polynomial follows its leading term alone. Degree parity and the sign of the leading coefficient set both ends.
Unit 1 · Topic 1.6
$$ \text{V.A. at } x = a \text{ if } q(a) = 0 \text{ and } p(a) \neq 0 $$
Vertical asymptote
A rational function has a vertical asymptote at a zero of the denominator that does not cancel with the numerator.
Unit 1 · Topic 1.9
$$ \text{hole where } (x - a) \text{ cancels} $$
Hole
A common factor of numerator and denominator produces a hole, not an asymptote. Its height comes from the reduced form.
Unit 1 · Topic 1.10
$$ \text{compare } \deg(p) \text{ and } \deg(q) $$
Horizontal asymptote
If the numerator degree is smaller, y = 0; if equal, the ratio of leading coefficients; if larger, there is no horizontal asymptote (a slant or polynomial end behavior instead).
Unit 1 · Topic 1.7
$$ g(x) = a\,f\!\big(b(x - h)\big) + k $$
Transformation form
Vertical stretch a, horizontal stretch 1/b, shift h right and k up. Factor b fully out of the input before reading the horizontal shift; f(2x + 6) is f(2(x + 3)).
Unit 1 · Topic 1.12
Unit 2
Exponential and Logarithmic Functions
Multiplicative change and how to undo it with logarithms.
$$ a_n = a_0 + d\,n $$
Arithmetic sequence
A constant common difference d added each step. Its graph over integer inputs is linear.
Unit 2 · Topic 2.1
$$ g_n = g_0 \cdot r^{\,n} $$
Geometric sequence
A constant common ratio r multiplied each step. Its graph over integer inputs is exponential.
Unit 2 · Topic 2.1
$$ f(x) = a \cdot b^{x}, \quad a = f(0), \; b = 1 + r $$
Exponential function
The initial value a is the output at x = 0 and the base b is the per-unit growth factor. A rate of change r gives b = 1 + r, so 6 percent growth is b = 1.06 and 6 percent decay is b = 0.94.
Unit 2 · Topic 2.3
$$ \log_b(c) = a \iff b^{a} = c $$
Definition of a logarithm
A logarithm is the exponent question: log base b of c is the power you raise b to in order to get c.
Unit 2 · Topic 2.9
$$ \log_b(xy) = \log_b x + \log_b y $$
Product, quotient, power rules
The log of a product is a sum, the log of a quotient is a difference, and log of x to the k is k times log x. There is no rule for the log of a SUM; log(x + y) does not split.
Unit 2 · Topic 2.12
$$ \log_b x = \dfrac{\log x}{\log b} $$
Change of base
Any logarithm can be rewritten as a quotient of logarithms in a common base. This is a quotient of logs, not the log of a quotient.
Unit 2 · Topic 2.10
$$ f\!\big(f^{-1}(x)\big) = x $$
Inverse function
An inverse undoes the function's process, so composing them returns the input. The notation f inverse does not mean 1 over f.
Unit 2 · Topic 2.8
Unit 3
Trigonometric and Polar Functions
Periodic behavior, the unit circle, and polar coordinates. Radians throughout.
At pi/6, pi/4, pi/3 the sine values are one half, root two over two, root three over two; cosine runs the same three in reverse. Signs follow the quadrant.
Unit 3 · Topic 3.3
$$ \sin^2\theta + \cos^2\theta = 1 $$
Pythagorean identity
The core identity, true for every angle. Rearranged, it gives sine from cosine or cosine from sine up to sign.
Each inverse trig function returns one angle from its restricted range. Solving an equation over a full period needs the second solution added by hand.
Unit 3 · Topic 3.9
$$ x = r\cos\theta, \; y = r\sin\theta, \; r^2 = x^2 + y^2 $$
Polar and rectangular
Convert between polar and rectangular with these relations. A negative r points opposite the theta ray, and polar coordinates are not unique.
Unit 3 · Topic 3.13
Unit 4
Functions Involving Parameters, Vectors, and Matrices
Parametric curves, vectors, and matrices. This unit is NOT assessed on the AP exam; schools include it based on local requirements.
$$ (x(t), \, y(t)) $$
Parametric function
One parameter drives both coordinates. The path has a direction and an extent set by the t interval, which eliminating the parameter throws away.
Unit 4 · Topic 4.1
$$ x = a + r\cos t, \; y = b + r\sin t $$
Parametric circle
A circle of radius r centered at (a, b). Direction and start are found by evaluating, not by reciting.
Unit 4 · Topic 4.4
$$ \big|\langle a, b\rangle\big| = \sqrt{a^2 + b^2} $$
Vector magnitude
Magnitude is the Pythagorean length of the components, never their sum. A vector is a displacement, not a location.
The dot product multiplies matching components and adds. The result is a single number, a scalar, not a vector.
Unit 4 · Topic 4.8
$$ \det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc $$
Determinant
The determinant is ad minus bc. The matrix is invertible exactly when the determinant is not zero; a zero determinant collapses the plane.
Unit 4 · Topic 4.11
$$ \text{columns} = \text{images of } (1,0) \text{ and } (0,1) $$
Linear transformation
The columns of a 2 by 2 matrix are where the basis vectors land. Read the transformation off its columns, and compose transformations right to left.
Unit 4 · Topic 4.12
Knowing the fact is not the same as not slipping on it
Most AP Precalculus points are lost to a small set of predictable misconceptions, not to forgotten formulas. Mistake Master diagnoses which ones are costing you points, then drills only those.
