AP Precalculus cheat sheet

A function assigns each input in its domain exactly one output; failing the vertical line test means the relation is not a function.

Average rate of change on [a, b] equals (f(b) − f(a))/(b − a), the slope of the secant line through the two graph points.

A function is even if f(−x) = f(x) for all x in its domain (graph symmetric about the y-axis); it is odd if f(−x) = −f(x) (symmetric about the origin).

Composition (f ∘ g)(x) = f(g(x)) requires x in the domain of g and g(x) in the domain of f; composition is generally not commutative.

Inverse functions satisfy f⁻¹(f(x)) = x and f(f⁻¹(x)) = x; the graph of f⁻¹ is the reflection of f across the line y = x.

A function has an inverse function only if it is one-to-one; restricting the domain of a non-one-to-one function can make it invertible.

For piecewise functions, evaluate each sub-expression on its own interval; domain restrictions come from denominators, even roots, and logarithms.

Transformations: g(x) = a·f(b(x − h)) + k scales vertically by a, compresses horizontally by 1/|b|, and shifts right h and up k.

End behavior of a polynomial follows its leading term aₙxⁿ: even n gives same-direction tails, odd n gives opposite-direction tails, and sign of aₙ sets the right-hand direction.

A zero of multiplicity m produces a graph that crosses the x-axis when m is odd and touches (is tangent) when m is even, flattening more as m grows.

Remainder Theorem: dividing p(x) by (x − c) leaves remainder p(c); Factor Theorem: (x − c) is a factor exactly when p(c) = 0.

Rational zeros of a polynomial with integer coefficients have the form p/q where p divides the constant term and q divides the leading coefficient.

A rational function has a vertical asymptote at x = c when the denominator is zero and does not cancel with a common factor in the numerator.

Horizontal asymptotes of N(x)/D(x) depend on degrees: numerator degree smaller gives y = 0, equal degrees give the ratio of leading coefficients, larger gives none.

A hole (removable discontinuity) occurs where a factor cancels from both numerator and denominator; the hole's x-value is excluded from the domain.

A rational inequality's sign changes only at zeros and at undefined points, so a sign chart on those critical values solves it.

Exponential growth/decay model f(t) = a·bᵗ has b > 1 for growth and 0 < b < 1 for decay; a is the initial value at t = 0.

For continuous compounding or natural growth, f(t) = a·e^{kt}, where k > 0 indicates growth and k < 0 indicates decay.

Logarithm definition: log_b(x) = y is equivalent to bʸ = x, with b > 0, b ≠ 1, and x > 0.

Product, quotient, and power rules: log_b(mn) = log_b m + log_b n; log_b(m/n) = log_b m − log_b n; log_b(mᵖ) = p·log_b m.

Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b), allowing evaluation with any available log key.

f(x) = bˣ and g(x) = log_b x are inverse functions, so their graphs reflect across y = x; the log graph has a vertical asymptote at x = 0.

To solve bᵘ = bᵛ, set exponents equal; to solve equations like a·bᵘ = c, isolate the exponential then take a logarithm of both sides.

Doubling time under exponential growth equals ln(2)/k, and half-life under decay equals ln(2)/|k| when written as a·e^{kt}.

On the unit circle a point at angle θ has coordinates (cos θ, sin θ), giving sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ.

Sine and cosine have period 2π; tangent has period π; for y = a·sin(b(θ − c)) + d the period is 2π/|b| and the amplitude is |a|.

y = a·sin(bθ + c) + d has phase shift −c/b and vertical shift d; the midline is y = d, with graph oscillating between d − |a| and d + |a|.

Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ; these are undefined where their base functions are zero.

Inverse trig functions are restricted to make them one-to-one: arcsin has range [−π/2, π/2], arccos has range [0, π], and arctan has range (−π/2, π/2).

Polar coordinates (r, θ) convert to rectangular via x = r·cos θ and y = r·sin θ; conversely r² = x² + y² and tan θ = y/x.

The polar curve r = a is a circle of radius |a| centered at the origin, while θ = k is a ray through the origin at angle k.

A polar equation can represent the same point with multiple pairs (r, θ), since (r, θ) and (r, θ + 2π) coincide and (−r, θ + π) also gives the same point.