Precalculus formula sheet
41 formulas · what each one means · a free lesson for each
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Functions
Composition
(f ∘ g)(x) = f(g(x))
Work out g first, then put the answer into f.Lesson: Composition and inverse functions →Inverse
f(f−1(x)) = x
Swap x and y, then solve for y. The graph is mirrored in y = x.Lesson: Composition and inverse functions →Average rate of change
f(b) − f(a)b − a
The slope between two points of the graph.Lesson: Average rate of change →Domain
no 0 at the bottom, nothing negative under an even root
The x values the function can use.Lesson: Domain and range →Moving graphs
Reflect
y = −f(x) y = f(−x)
−f(x) flips over the x-axis; f(−x) flips over the y-axis.Lesson: Graph reflections →Vectors
Vector from two points
v = ⟨x2 − x1, y2 − y1⟩
End minus start.Lesson: Vector from two points / magnitude →Magnitude
‖v‖ = √a2 + b2
The length of ⟨a, b⟩: Pythagoras.Lesson: Vector from two points / magnitude →Direction
tan θ = ba
Check the quadrant of ⟨a, b⟩.Lesson: Magnitude and direction, unit vector →Add and scale
⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩ k⟨a, b⟩ = ⟨ka, kb⟩
Lesson: Adding, subtracting, × a number →Dot product
u • v = ac + bd
For ⟨a, b⟩ and ⟨c, d⟩. 0 means they are at 90°.Lesson: Dot product, angle between two vectors →Angle between vectors
cos θ = u • v‖u‖ × ‖v‖
Lesson: Dot product, angle between two vectors →Conic sections
Parabola
(x − h)2 = 4p(y − k)
Vertex (h, k); focus p from the vertex, directrix p on the other side.Lesson: Parabola →Ellipse
(x − h)2a2 + (y − k)2b2 = 1, c2 = a2 − b2
a is the long half-axis; foci are c from the center.Lesson: Ellipse →Hyperbola
(x − h)2a2 − (y − k)2b2 = 1, c2 = a2 + b2
Asymptotes y − k = ±(b/a)(x − h).Lesson: Hyperbola →Polar and parametric
Polar form of a complex number
z = r(cos θ + i sin θ)
r = |z| = √(a² + b²).Lesson: Polar form of complex numbers, De Moivre's theorem →De Moivre's theorem
zn = rn(cos nθ + i sin nθ)
Power the length, multiply the angle.Lesson: Polar form of complex numbers, De Moivre's theorem →Parametric equations
x = f(t), y = g(t)
Get rid of t to find the x-y equation.Lesson: Parametric equations →Sequences and series
Arithmetic series
Sn = n(a1 + an)2
Number of terms × the mean of the first and last.Lesson: Series →Sigma notation
Σni=1 ai = a1 + a2 + … + an
Add the terms from i = 1 to n.Lesson: Sigma notation →Binomial theorem
(a + b)n = Σnk=0 (nk) an−k bk
The numbers in front come from Pascal's triangle.Lesson: The binomial theorem →Log rules
Definition
logb x = y ⇔ by = x
A log is a power: the power of b that gives x.Lesson: Exponential equations and logarithms →Product rule
logb(MN) = logb M + logb N
Times inside → plus outside.Lesson: Log properties with letters →Quotient rule
logb MN = logb M − logb N
Divide inside → minus outside.Lesson: Log properties with letters →Change of base
logb x = log xlog b = ln xln b
How a calculator finds any log.Lesson: Exponential equations and logarithms →Log equations
check every answer: no log of 0 or a negative
Some answers are false and must be thrown away.Lesson: Log equations with false answers →More formula sheets
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