Pre-Calculus Reference Sheet

The set of all possible input (x) values

The set of all possible output (y) values

A function consisting of terms with non-negative integer exponents (e.g., f(x) = 3x² + 2x - 1)

The highest power of the variable in the polynomial

For 3x⁴ + 2x - 1, the degree is 4

The degree (even/odd) and the sign of the leading coefficient

A function that is the ratio of two polynomials, f(x) = p(x)/q(x)

A vertical line x = a where the function approaches infinity, occurring where the denominator equals zero

Set the denominator equal to zero

A horizontal line y = b that the function approaches as x approaches positive or negative infinity

A function of the form f(x) = a·bˣ where b > 0 and b ≠ 1

A logarithmic function

2.71828...

Named after Euler

log(ab) = log(a) + log(b)

log(a/b) = log(a) - log(b)

log(aⁿ) = n · log(a)

(f ∘ g)(x) = f(g(x)) — applying one function to the result of another

Swap x and y, then solve for y

360 degrees

2π radians

Radians = Degrees × (π/180)

(1, 0)

(0, 1)

1/2

One of the special angle values

1/2

√2/2

2π

3

The coefficient in front of sin

Quadrants I and II

Remember: All Students Take Calculus

Quadrants I and IV

sin(θ) = cos(90° - θ) — they are cofunctions

tan(θ) = sin(θ)/cos(θ)

π

csc (cosecant), sec (secant), cot (cotangent)

sin²(θ) + cos²(θ) = 1

1 + tan²(θ) = sec²(θ)

Divide the first Pythagorean identity by cos²(θ)

sin(2θ) = 2sin(θ)cos(θ)

cos(2θ) = cos²(θ) - sin²(θ)

There are three equivalent forms

sin(A)cos(B) + cos(A)sin(B)

cos(A)cos(B) - sin(A)sin(B)

a/sin(A) = b/sin(B) = c/sin(C)

c² = a² + b² - 2ab·cos(C)

A generalization of the Pythagorean theorem

±√((1 - cos(θ))/2)

Area = (1/2)ab·sin(C), where a and b are sides and C is the included angle

A sequence where each term differs from the previous by a constant (common difference)

aₙ = a₁ + (n - 1)d

d is the common difference

A sequence where each term is multiplied by a constant (common ratio)

aₙ = a₁ · rⁿ⁻¹

r is the common ratio

Sₙ = n(a₁ + aₙ)/2

Sₙ = a₁(1 - rⁿ)/(1 - r), where r ≠ 1

When the absolute value of the common ratio is less than 1 (|r| < 1)

S = a₁/(1 - r)

Expanding expressions of the form (a + b)ⁿ

A compact way to write the sum of a series using an index variable

The value a function approaches as the input approaches a particular value

The function approaches different values from the left and right, or approaches infinity

0

A function with no breaks, holes, or jumps in its graph — the limit equals the function value at every point

1

A famous limit used in calculus

An expression like 0/0 or ∞/∞ where the limit cannot be directly evaluated

A 'hole' in the graph where the limit exists but the function is either undefined or has a different value

If f is continuous on [a,b], then f takes every value between f(a) and f(b)

A coordinate system using distance from origin (r) and angle from positive x-axis (θ)

x = r·cos(θ)

y = r·sin(θ)

r = √(x² + y²)

r = a (constant)

A cardioid

A rose curve with n petals (if n is odd) or 2n petals (if n is even)

Equations where x and y are both expressed as functions of a third variable (parameter), usually t

x = r·cos(t), y = r·sin(t)

Solve one parametric equation for t, then substitute into the other

x = v₀cos(θ)·t, y = v₀sin(θ)·t - (1/2)gt²

Determined by increasing values of the parameter t