Formula and reference support

f(a) means substitute x = a

For f: x ↦ y, x is the input and y is the output.

f(g(x)) means apply g first, then f.

To find f⁻¹(x), write y = f(x), swap x and y, then rearrange

Check f(f⁻¹(x)) = x

y = f(x) + a moves the graph up a

y = f(x − a) moves the graph right a

y = −f(x) reflects in the x-axis

m = y₂ − y₁x₂ − x₁

y − b = m(x − a)

Parallel lines have equal gradients; perpendicular gradients multiply to −1.

Distance = √((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint = (x₁ + x₂2, y₁ + y₂2)

(x − a)² + (y − b)² = r²

x = (-b ± √(b² − 4ac))/(2a)

Discriminant = b² − 4ac

Complete the square to reveal the turning point.

If f(a) = 0, then (x − a) is a factor.

When f(x) is divided by (x − a), the remainder is f(a).

d/dx(xⁿ) = nxⁿ⁻¹

d/dx(axⁿ) = anxⁿ⁻¹

Constants differentiate to 0.

Differentiate to find dy/dx

Substitute the x-coordinate for the tangent gradient.

Normal gradient = −1m

Use y − b = m(x − a).

Stationary point: dy/dx = 0

d²y/dx² > 0 gives a minimum; d²y/dx² < 0 gives a maximum

For optimisation, interpret the value in context.

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1

∫k dx = kx + C

∫ from a to b f(x) dx = F(b) − F(a)

Area under a positive curve equals the definite integral.

Signed area below the x-axis is negative.

sin 30° = 12, cos 60° = 12

sin 45° = cos 45° = √22

sin² x + cos² x = 1

Find the reference angle.

Use CAST or the graph to find all solutions.

Check the stated interval.

Amplitude controls height.

Period controls repeat length.

Phase shift moves the graph horizontally.

aˣ = b is equivalent to logₐ b = x.

Exponential and logarithmic functions are inverses when bases match.

log(ab) = log a + log b

log(ab) = log a − log b

log(aⁿ) = n log a

Log arguments must be positive.

If a = (p, q), then |a| = √(p² + q²).

ka = (kp, kq)

a + b = (a₁ + b₁, a₂ + b₂).

a · b = a₁b₁ + a₂b₂ + a₃b₃.

a · b = |a||b|cos θ.

If a · b = 0, non-zero vectors are perpendicular.

Use the previous term to generate the next term.

For a limit L, substitute L into both sides where appropriate.

Check convergence numerically before claiming a limit.