AP Precalculus — Cheatsheet
Formulas, exam-day tips, and key terms on one page.
Formulas & relationships
End-behavior rules
even degree → arms agree · odd degree → arms disagree · sign of lead sets the right arm
Positive leading coefficient: right arm rises (f → +∞ as x → +∞). Negative leading coefficient: right arm falls.
Degree = total zeros with multiplicity
deg f = sum of the multiplicities of all zeros
Over the complex numbers a degree-n polynomial has exactly n zeros counted with multiplicity. Adding the exponents of the factors recovers the degree.
Horizontal-asymptote test
deg(top) < deg(bottom) → y = 0 · equal → y = (lead top)/(lead bottom) · top > bottom → none
This only describes end behavior. The graph may cross a horizontal asymptote in the middle; the rule is about x → ±∞.
Transformation summary
g(x) = a·f(x − h) + k
h shifts horizontally (right for −h inside), k shifts vertically, a scales/reflects vertically (a < 0 flips across the x-axis). A minus on x inside, f(−x), reflects across the y-axis.
Exponential model
f(x) = a · bˣ
a = value at x = 0 (initial amount). b > 1 grows, 0 < b < 1 decays. The percent change per step is (b − 1)·100%.
Logarithm laws
log(xy) = log x + log y · log(x/y) = log x − log y · log(xⁿ) = n·log x
Products become sums, quotients become differences, and exponents come out front as multipliers. Note that log(x + y) does NOT simplify.
Inverse composition identity
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Composing a function with its inverse (in either order, on the appropriate domain) returns the original input. For example, 2^(log₂ x) = x.
Solving bˣ = c
x = log(c) / log(b)
Take the logarithm of both sides and divide. Any base works (common log or natural log) as long as you use it on both sides.
Point on the unit circle
(x, y) = (cos θ, sin θ)
Quadrantal angles: θ = 0 → (1, 0); θ = π/2 → (0, 1); θ = π → (−1, 0); θ = 3π/2 → (0, −1).
Core identities
sin²θ + cos²θ = 1 · tan θ = sin θ / cos θ · cot θ = cos θ / sin θ
Dividing the Pythagorean identity by cos²θ gives tan²θ + 1 = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.
Sinusoid parameters
y = A sin(Bx) + D · amplitude = |A| · period = 2π/B · midline y = D
Amplitude is a distance (never negative). The period comes only from B; the amplitude A and midline D do not affect it.
Coordinate conversions
x = r cos θ · y = r sin θ · r = √(x² + y²) · tan θ = y/x
The first pair converts polar to rectangular; the second pair converts rectangular to polar. Always check the quadrant when finding θ.
Parametric form
x = f(t) · y = g(t)
Evaluate a point by plugging in a value of t. Eliminate t by solving one equation for it and substituting into the other.
Vector operations
⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩ · |⟨a, b⟩| = √(a² + b²)
Addition is component-wise; magnitude is the Pythagorean length of the components. Scalar multiplication k⟨a, b⟩ = ⟨ka, kb⟩ scales the length by |k|.
Determinant and product size
det[[a, b], [c, d]] = ad − bc · (m×n)(n×p) = m×p
The determinant is the main diagonal product minus the anti-diagonal product. Matrix multiplication requires the first matrix’s column count to equal the second’s row count.
Common 2×2 transformations
reflect x-axis [[1, 0], [0, −1]] · reflect y-axis [[−1, 0], [0, 1]] · identity [[1, 0], [0, 1]]
Each column shows where a basis vector goes. The identity leaves every vector fixed; a reflection flips the sign of one coordinate.
On the exam
- On the AP exam, state end behavior with limit-style language: “as x → ∞, f(x) → ∞.” Naming the degree parity and the sign of the leading coefficient is the reasoning that earns the point.
- When a graph is given, read multiplicity from the shape: a straight-through crossing is multiplicity 1, a flattened crossing is multiplicity 3+, and a bounce is even. This lets you rebuild the factored form directly from a picture.
- Always factor fully first. The exam loves a rational function whose denominator “looks like” two asymptotes but where one factor cancels into a hole. Simplify, then read vertical asymptotes and holes off the reduced form.
- When several transformations combine, handle inside changes (horizontal shifts, y-axis flips) separately from outside changes (vertical shifts, x-axis flips). Describe each precisely — the AP rubric awards the direction and the axis, not just the word “shift.”
- For growth described by a doubling/halving time T, use base 2 (or 1/2) with exponent t/T. For a percent rate r per period, use base (1 + r). Matching the base and the exponent to the description is the graded step.
- Use the power rule to solve exponential equations: log(bˣ) = x·log b lets you pull the variable out of the exponent. Recognizing when to expand versus condense is what the free-response rubric rewards.
- Use the composition identity to simplify: e^(ln x) = x and log(10ˣ) = x. Spotting a function next to its inverse lets you collapse a messy expression in one step.
- Pick the log base that simplifies the work: natural log (ln) pairs with base e, common log with base 10, but log₃ 20 = log 20 / log 3 works for any base via the change-of-base formula. State the exact answer before rounding.
- Sign by quadrant: in QII cosine is negative and sine positive; in QIII both are negative; in QIV cosine is positive and sine negative. Combine a reference angle with the quadrant’s sign to get any exact value.
- When a proof stalls, convert every function to sine and cosine and look for the Pythagorean identity. Most AP simplification problems collapse once everything is written over a common sin/cos foundation.
- From a graph, read the midline and amplitude off the max and min: midline D = (max + min)/2 and amplitude A = (max − min)/2. Then find the period by measuring one full cycle and solve B = 2π/period.
- When converting rectangular → polar, arctan alone can give the wrong quadrant, since it only returns angles in (−π/2, π/2). Sketch the point first and add π when it lies in the second or third quadrant.
- After eliminating the parameter, note any restriction on the resulting curve. If t only ranges over part of an interval, the parametric curve may be just a piece of the full graph — the AP rubric checks that you respect the domain.
- To decompose a vector of magnitude r at angle θ, use components ⟨r cos θ, r sin θ⟩ — the same conversion as polar coordinates. This links force/velocity problems back to unit-circle trigonometry.
- Before multiplying, write the dimensions side by side, (2×3)(3×2), and check that the inner numbers match. The outer numbers give the answer’s size — a fast way to catch an impossible product on the exam.
- To find the matrix of a described transformation, track where ⟨1, 0⟩ and ⟨0, 1⟩ land and place those images as columns. This construction handles rotations, reflections, and stretches uniformly on the AP exam.
How to get a 5
- Always check for extraneous solutions.
- Know your unit circle cold.
Key terms
Logarithm Base Change Formula — log_b(x) = ln(x) / ln(b)
Compound Interest Formulas — Periodic: A = P(1 + r/n)^(nt). Continuous: A = Pe^(rt).
End Behavior of Polynomials — Determined by the leading term ax^n.
Rational Functions: Horizontal Asymptotes — Compare degrees of numerator and denominator.
Even vs. Odd Functions — Even: f(-x) = f(x). Odd: f(-x) = -f(x).
Trig Identities: Pythagorean — sin²θ + cos²θ = 1