Full Formula Sheet — Units 1–8

Unit circle equation:

All points \((x,y)\) on the unit circle satisfy this equation. Radius \(= 1\).

Degrees & Radians

• $$360^\circ = 2\pi \text{ radians}$$
• $$1^\circ = \frac{\pi}{180} \text{ radians}$$
• $$1 \text{ radian} = \frac{180^\circ}{\pi}$$

Key identities

$$\sin^2\theta + \cos^2\theta = 1$$

Basic trig functions

• $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$
• $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
• $$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

Master form & parameters

Master form: \(y = a\) · \(f(b(x + c)) + d\)

• a: vertical stretch/compression by \(|a|\); reflect across \(x\)-axis if \(a\) < \(0\)
• b: horizontal stretch/compression (scale \(= 1/|b\)); reflect across \(y\)-axis if \(b\) < \(0\)
• c: horizontal shift (note sign inside)
• d: vertical shift

Mapping rule

If \((x,y)\) is on \(y=f(x)\), then on \(y=a f(b(x+c))+d\) the corresponding point is:

\(x' = x/b - c\),   \(y' = a y + d\)

Radical functions

Parent functions: \(y = \sqrt{x}\) (domain \(x ≥ 0\)) and \(y = \sqrt[3]{x})\) (all reals).

General transform: \(y = a \sqrt{b(x+c)} + d\)

Rational functions

\(R(x) = P(x)/Q(x)\). Domain excludes roots of \(Q\). Vertical asymptotes where denominator zeros remain after cancellation; holes where factors cancel.

Horizontal asymptotes depend on degrees of numerator/denominator.

Exponential functions

General: \(y = a b^{kx} + d\). Domain: all reals. Horizontal asymptote: \(y = d\).

Logarithms

Definition: \(log_b(N)=e\) iff \(b^e=N\). Domain: \((0,∞)\). Properties: product, quotient, power, change of base.

Core facts

• Polynomial: \(f(x)=a_n x^n + ... + a_0\). Degree \(= n.\)
• End behaviour determined by degree parity and leading coefficient.
• Multiplicity: odd \(\)= cross, even \(=\) bounce.

Graphing checklist

1. Factor, find zeros and multiplicities.
2. Find \(y\)-intercept \(f(0)\).
3. Sketch end behaviour and turning points \((≤ n−1)\).

Factorials & counting

• \(n! = n·(n−1)·...·1\), and \(0! = 1\)
• \(P(n,k) = n!/(n−k)!\), \(C(n,k) = n!/(k!(n−k)!)\)

Binomial theorem

\((a+b)^n = Σ_{k=0}^n C_{(n,k)} a^{n−k} b^k\)

Addition & subtraction

• \(sin(A±B) = sinA cosB ± cosA sinB\)
• \(cos(A±B) = cosA cosB ∓ sinA sinB\)
• \(tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)\)

Double-angle

• \(sin2a = 2 sin a cos a\)
• \(cos2a = cos^2 a − sin^2 a = 2 cos^2 a − 1 = 1 − 2 sin^2 a\)

Notation & operations

• \((f+g)(x) = f(x)+g(x)\), \((f·g)(x)=f(x)g(x)\), \((f/g)(x)=f(x)/g(x)\) where \(g(x)≠0\)
• \((f∘g)(x)=f(g(x))\). Domain caution: \(g(x)\) must be in domain\((f)\).

Domains

Domain\((f+g)\)=Domain\((f)\) ∩ Domain\((g)\). For composition use { \(x\) in Domain\((g)\): \(g(x)\) in Domain\((f)\) }.