Precalculus
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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.
Average rate of change
AROC on [a, b] = [f(b) − f(a)] / (b − a)
Units are (units of f) per (unit of x). Writing the units is half the point on an AP free-response question.
Concavity from rates of change
rate of change increasing → concave up · rate of change decreasing → concave down
True regardless of whether the function itself is increasing or decreasing. A falling graph can be concave up (falling ever more gently) or concave down (falling ever faster).
Division algorithm for polynomials
p(x) = d(x)·q(x) + r(x), with deg r < deg d — equivalently p(x)/d(x) = q(x) + r(x)/d(x)
The second form is the one used to find slant asymptotes and to rewrite rational functions.
Remainder and Factor Theorems
p(x) ÷ (x − a) leaves remainder p(a) · and (x − a) is a factor ⟺ p(a) = 0
Watch the sign: dividing by (x + 3) means a = −3, so the relevant value is p(−3).
Fundamental Theorem of Algebra
degree n ⟹ exactly n complex zeros counted with multiplicity
x² − 6x + 9 = (x − 3)² has "two" zeros: 3 twice. Multiplicity is what makes the count exact.
Conjugate pair theorem
real coefficients and p(a + bi) = 0 ⟹ p(a − bi) = 0
A conjugate pair multiplies to the real quadratic (x − a)² + b², which is why real polynomials factor into real linear and real quadratic pieces.
Sign chart method
find all zeros → mark them on a number line → test one point per interval → read off the intervals with the sign you want
For rational expressions, mark the zeros of the numerator AND the zeros of the denominator. Both are places the sign can flip.
Hole vs vertical asymptote
factor cancels → hole at that x · factor remains in denominator → vertical asymptote
A factor appearing twice in the denominator and once in the numerator still leaves one copy behind — so it is an asymptote, not a hole.
End behavior of a rational function
n < d → y = 0 · n = d → y = (lead of numerator)/(lead of denominator) · n = d + 1 → slant asymptote · n > d + 1 → grows like a power
n and d are the degrees of numerator and denominator. Only the n = d + 1 case gives a line.
Composition and its domain
(f∘g)(x) = f(g(x)) — defined only when x is in the domain of g AND g(x) is in the domain of f
Both conditions. The second is the one that gets dropped, and dropping it is the standard error.
Parity tests
even: f(−x) = f(x) · odd: f(−x) = −f(x) · neither: some x where both fail
The names come from powers: xⁿ is even when n is even and odd when n is odd. Testing means substituting −x and simplifying, then comparing to f(x) and to −f(x).
Combining parities
product: even·even = even, odd·odd = even, even·odd = odd · sum: like + like keeps the type, mixed gives neither
Composition too: f∘g is even whenever g is even, whatever f is — because g(−x) = g(x) means the composition never sees the sign.
Identifying a model from a table
constant 1st differences → linear · constant 2nd → quadratic · constant nth → degree n · constant ratios → exponential · periodic → sinusoidal
Requires equally spaced inputs. With unequal spacing, differences and ratios are not comparable and the test does not apply.
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.
Explicit formulas
arithmetic: aₙ = a₁ + (n − 1)d · geometric: gₙ = g₁ · r^(n−1)
The exponent and the multiplier are (n − 1), not n, when indexing from a₁. Off-by-one here is the most common error in the topic.
Change of base
log_b(x) = ln(x)/ln(b) = log(x)/log(b) — any common base works
This is a ratio, not a difference. log(x)/log(b) is not log(x) − log(b), and confusing the two is the standard error.
Compound interest, periodic and continuous
A = P(1 + r/n)^(nt) · A = P·e^(rt) as n → ∞
r is the nominal annual rate as a decimal, n the compoundings per year, t the years. The second formula is the first one's limit, not a different model.
Converting between base and continuous rate
b^t = e^(kt) where k = ln b · equivalently b = e^k
k > 0 means growth (b > 1), k < 0 means decay (0 < b < 1). k is often called the continuous rate or the relative growth rate.
Half-life and doubling-time models
A(t) = A₀·(1/2)^(t/h) with half-life h · P(t) = P₀·2^(t/T) with doubling time T
The exponent t/h counts *how many half-lives have elapsed*. That reading makes the formula reconstructable from scratch.
Doubling time
T = ln 2 / ln(1 + r) for periodic growth · T = ln 2 / k ≈ 0.693/k for continuous growth
The familiar "rule of 70" is this formula in disguise: 100·ln 2 ≈ 69.3, so T ≈ 70/(percent rate) is a first-order approximation, good to a few percent for small rates.
Linearizing an exponential
y = a·bˣ ⟹ log y = (log b)·x + log a
Slope = log b, so b = 10^slope. Intercept = log a, so a = 10^intercept. Using ln instead, b = e^slope and a = e^intercept.
Which plot straightens which family
exponential y = a·bˣ → semi-log is linear · power law y = a·xⁿ → log-log is linear, with slope n
A quick discriminator: exponentials eventually beat every power law, and only the exponential has a constant ratio over equal x-steps.
Three logarithmic scales
pH = −log[H⁺] · dB = 10·log(I/I₀) · magnitude difference: I₁/I₂ = 10^(M₁ − M₂)
The minus sign in pH exists so that ordinary concentrations give positive pH values. I₀ for decibels is the threshold of hearing, 10⁻¹² W/m².
The three log rules used to condense
log a + log b = log(ab) · log a − log b = log(a/b) · n·log a = log(aⁿ)
Every one of these requires a > 0 and b > 0. The rules are identities only on the domain where both sides are defined — which is precisely why extraneous solutions appear.
Transformed asymptotes, domains and ranges
y = a·b^(x−h) + k → asymptote y = k, domain all reals, range y > k (if a > 0) · y = a·log_b(x − h) + k → asymptote x = h, domain x > h, range all reals
For a < 0 the exponential's range flips to y < k. The reflection is in the sign of a, not in k.
The log product rule as a transformation identity
log_b(cx) = log_b(c) + log_b(x) — a horizontal scaling equals a vertical shift
Similarly log_b(xⁿ) = n·log_b(x): raising the input to a power is a vertical stretch.
Logistic model
P(t) = L / (1 + a·e^(−kt)), with carrying capacity L
As t → ∞ the term a·e^(−kt) → 0 and P → L. As t → −∞ that term dominates and P → 0. So the curve runs from 0 up to L.
Inflection point of a logistic curve
occurs at P = L/2, at time t = ln(a)/k
Setting L/(1 + a·e^(−kt)) = L/2 requires a·e^(−kt) = 1, giving e^(kt) = a and t = ln(a)/k.
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.
Radian conversions and arc length
π radians = 180° · s = rθ · A = ½r²θ (θ in radians)
Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. Both arc-length formulas fail outright if θ is in degrees.
Angular and linear speed
ω = θ/t · v = rω · one revolution = 2π radians
Convert revolutions per minute to radians per second by multiplying by 2π/60. ω must be in radians per unit time for v = rω to hold.
The standard sinusoid
y = a·sin(b(x − h)) + k · amplitude = |a| · period = 2π/|b| · midline y = k · phase shift h
The equation must be factored as b(x − h) before h can be read. In y = sin(2x − π), the shift is not π; factoring gives sin(2(x − π/2)), so h = π/2.
Asymptotes and periods
tan, sec: asymptotes at x = π/2 + nπ · cot, csc: asymptotes at x = nπ · period of tan and cot is π · period of sec and csc is 2π
n ranges over all integers. Tangent and cotangent are the only two with period π.
Period of a transformed tangent
y = a·tan(b(x − h)) + k has period π/|b|, not 2π/|b|
The single most common error in this topic. Tangent and cotangent use π in the numerator; the other four use 2π.
Domains and ranges of the inverses
arcsin: domain [−1, 1], range [−π/2, π/2] · arccos: domain [−1, 1], range [0, π] · arctan: domain all reals, range (−π/2, π/2)
arcsin and arctan return values in quadrants I and IV (so negative inputs give negative angles). arccos returns quadrants I and II (so it never returns a negative angle).
Generating all solutions
sin x = c → x = α and π − α, plus 2πn · cos x = c → x = α and −α (or 2π − α), plus 2πn · tan x = c → x = α, plus πn
α is the value the inverse function returns. Tangent needs only one base solution per period because its period is π, not 2π.
Sum and difference formulas
sin(A ± B) = sin A cos B ± cos A sin B · cos(A ± B) = cos A cos B ∓ sin A sin B
Note the sign reversal in the cosine formula: cos(A + B) uses a **minus**. This is the detail worth over-rehearsing.
Double-angle formulas
sin 2A = 2 sin A cos A · cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
Pick the cos 2A form that matches what you already have. If the expression contains sin²A, use 1 − 2sin²A; if it contains cos²A, use 2cos²A − 1.
The two laws
a/sin A = b/sin B = c/sin C · c² = a² + b² − 2ab·cos C
In the Law of Cosines, the angle C must be the one *opposite* the side c. With C = 90° the cosine term vanishes and it reduces to the Pythagorean theorem.
Resolving SSA
find the acute angle α from the Law of Sines, then test 180° − α · the supplement is valid only if the three angles still sum to 180°
If the side opposite the given angle is shorter than the height a·sin A, no triangle exists — the arc cannot reach the base.
Area from two sides and the included angle
Area = ½ab·sin C
C must be the angle *between* the sides a and b. With C = 90°, sin C = 1 and this reduces to ½(base)(height).
Polar curve families
r = a: circle radius a · r = a cos θ: circle diameter a on the polar axis · r = a cos(nθ): rose · r = a ± b sin θ: limaçon, cardioid when a = b
r = a cos θ is a circle of *radius* a/2 centered at (a/2, 0), not a circle of radius a. The a is the diameter.
Petal count and length
r = a·cos(nθ): n odd → n petals · n even → 2n petals · petal length |a|
A cosine rose has a petal along the polar axis; a sine rose is the same curve rotated. The count rule is identical for both.
Average rate of change of a polar function
AROC of r on [θ₁, θ₂] = [f(θ₂) − f(θ₁)] / (θ₂ − θ₁)
Units are distance per radian. A positive value means the curve is receding from the origin on average over that interval.
Estimating a sinusoidal fit
k = (max + min)/2 · a = (max − min)/2 · b = 2π/period · h = x-value of a maximum (for a cosine model)
Real data rarely has an exact maximum at a sampled point, so h is an estimate. Averaging two consecutive cycles improves it.
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.
The four conic sections in standard form
circle: (x − h)² + (y − k)² = r² · ellipse: (x − h)²/a² + (y − k)²/b² = 1 · parabola: (y − k) = a(x − h)² · hyperbola: (x − h)²/a² − (y − k)²/b² = 1
The distinguishing feature is how the squared terms combine: both positive and equal coefficients gives a circle, both positive and unequal gives an ellipse, opposite signs gives a hyperbola, and only one squared term gives a parabola.
Parametrizing a circle and an ellipse
circle radius r: x = r cos t, y = r sin t · ellipse: x = a cos t, y = b sin t
Both are traced counterclockwise starting at (r, 0) or (a, 0). Swapping sine and cosine, or negating one, reverses or rotates the starting point.
Two formulas for the dot product
u · v = u₁v₁ + u₂v₂ · u · v = |u||v|cos θ
Setting the two equal and solving for cos θ is how the angle between vectors is found. Note that the dot product is commutative: u · v = v · u.
Angle between vectors, and the perpendicularity test
cos θ = (u · v)/(|u||v|) · u ⟂ v ⟺ u · v = 0
Compute the two magnitudes and the dot product, divide, then take arccos. The result always lies in [0°, 180°], since arccos does.
Determinant and inverse of a 2 × 2 matrix
for A = [[a, b], [c, d]]: det A = ad − bc · A⁻¹ = (1/det A)·[[d, −b], [−c, a]]
Swap the diagonal entries, negate the off-diagonal ones, divide by the determinant. The inverse exists exactly when det A ≠ 0.
Determinant and solution count
det A ≠ 0 → exactly one solution · det A = 0 → either no solution or infinitely many
The determinant also measures area scaling: |det A| is the factor by which A scales areas, and det A = 0 means it collapses the plane onto a line.
Transition matrix and projection
T = [[stay_city, from_suburb], [to_suburb, stay_suburb]] · next = T · current · after n steps: Tⁿ · initial
Each **column** must sum to 1 in this convention: everyone currently in a state has to go somewhere. Checking the column sums catches setup errors immediately.

On the exam

How to get a 5

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
Average rate of change of f on [a, b] — (f(b) − f(a))/(b − a) — the slope of the secant line through the endpoints.
Multiplicity of a zero (graph behavior) — Odd multiplicity: the graph crosses the x-axis. Even multiplicity: the graph touches and turns around.
End behavior of a polynomial — Determined entirely by the leading term: its degree (odd/even) and the sign of its leading coefficient.
Vertical asymptote vs. hole — Both come from denominator zeros. If the factor cancels with the numerator, it is a hole; if it survives, it is a vertical asymptote.
Horizontal asymptote rules for rational functions — Compare degrees: top < bottom gives y = 0; equal degrees give the ratio of leading coefficients; top > bottom gives no horizontal asymptote (possibly a slant).
Slant (oblique) asymptote — Occurs when the numerator’s degree exceeds the denominator’s by exactly 1; found by polynomial long division (quotient part).
Inverse function — Reflects the graph across y = x and swaps domain and range. Exists only if the original function is one-to-one, which the horizontal line test decides.
Definition of a logarithm — log_b(x) = y means b^y = x. A log is an exponent.
Log properties — log(MN) = log M + log N; log(M/N) = log M − log N; log(M^k) = k log M.
Change of base formula — log_b(x) = log(x)/log(b), using any consistent base. Needed because calculators offer only base 10 and base e.
Half-life model — A(t) = A₀(1/2)^(t/h), where h is the half-life. Every h units of time, the amount halves.
Continuous growth model — A(t) = A₀e^(kt); k > 0 grows, k < 0 decays. Doubling time = ln(2)/k.
Semi-log plot — Plotting log(y) against x linearises exponential data. A straight line on a semi-log plot is evidence the underlying model is exponential.
Amplitude, midline, period of a sinusoid — For y = A sin(B(x − C)) + D: amplitude |A|, midline y = D, period 2π/B, phase shift C.
Unit-circle values to memorize — sin and cos at 0, π/6, π/4, π/3, π/2 (0, 1/2, √2/2, √3/2, 1 pattern) plus quadrant signs (ASTC).
Pythagorean identity — sin²θ + cos²θ = 1, with tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ following by division.
Period of tangent — π (not 2π). Tangent has vertical asymptotes where cos θ = 0, at odd multiples of π/2.
Polar-to-rectangular conversion — x = r cos θ, y = r sin θ; also r² = x² + y² and tan θ = y/x.