Pure Maths

Transformation y = |f(x)|

Negative values of y are reflected in x-axis

Transformation y = f(x) + a

+a in y-axis

Transformation y = f(x + a)

-a in x-axis

Transformation y = af(x)

All y-coordinates ×a

Transformation y = f(ax)

All x-coordinates ÷a

Transformation y = -f(x)

Reflection in x-axis

Transformation y = f(-x)

Reflection in y-axis

When is a binomial expansion valid (yr13)?

For (a + bx)ⁿ, valid when | bx/a | < 1

or |x| < a/b

x = aⁿ as a log?

n = log_ax

log_a (x^k) rearranged

klog_ax

log(xy)

log(x) + log(y)

log(x/y)

log(x) - log(y)

Order of transformations

x: reflect, stretch, translate | y: translate, stretch, reflect , modulus

π Radians in degrees

180

Condition for increasing function?

dy/dx ≥ 0

Condition for decreasing function?

dy/dx ≤ 0

Condition for local minimum?

d²y/dx² > 0

Condition for local maximum?

d²y/dx² < 0

Condition for concave function?

d²y/dx² ≤ 0

Condition for convex function?

d²y/dx² ≥ 0

Percentage Change

Area of circle segment

How to integrate a parametric function?

Differentiate

y = ln(5x)

dy/dx = 1/x

Graph of sec x

Graph of cosec x

Graph of cot x

Domain and range of sec x

Domain: x ∈ R, x ≠ any odd multiple of 90

Range: between -1 and 1

Domain and range of cosec x

Domain: x ∈ R, x ≠ any multiple of 180

Range: between -1 and 1

Domain and range of cot x

Domain: x ∈ R, x ≠ any multiple of 180

Range: y ∈ R

Differentiate y = a^kx

dy/dx = a^kx k ln(a)

nth term of a geometric sequence

ar^n-1

Equation for length of a line

[ (x₁ + x₂)² + (y₁ + y₂)² ]^1/2

Arc Length equation

arc length = 2πr × (θ/360)

if in radians then (θ/2π)