Formulae: learn it, or look it up?

• Dividing: $a^x÷a^y=a^{x-y}$
• Power of a power: ${\left(a^x\right)}^y=a^{xy}$

a

base (positive for non-integer powers)

x, y

indices

Same base only: add indices to multiply, subtract to divide. Never add the bases.

• Unit fraction: $a^{\frac{1}{n}}=\sqrt[n]{a}$
• Fraction: $a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m$
• Zero index: $a^0=1$

a

positive base, a > 0

m, n

integers, n > 0

Take the root first, then the power, to keep the numbers small: 8 to the power ⅔ is (∛8)² = 2² = 4.

• Dividing: $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
• Square of a surd: $\sqrt{a}\sqrt{a}=a$

a, b

positive numbers, a > 0 and b > 0

√(a + b) is not √a + √b. Simplify by taking out the largest square factor: √48 = 4√3.

• Single surd: $\frac{1}{\sqrt{b}}=\frac{\sqrt{b}}{b},b>0$

a

rational number

b

positive integer, not a square

Multiply top and bottom by the conjugate (change the sign between the terms).

x

the unknown

a, b, c

coefficients of ax² + bx + c = 0, a ≠ 0

Rearrange to = 0 first. Use it on equations in a function of x too, e.g. a quadratic in eˣ or sin x.

• Equal roots: $b^2-4ac=0\text{ one repeated root}$
• No real roots: $b^2-4ac<0\text{ no real roots}$

a, b, c

coefficients of ax² + bx + c, a ≠ 0

'Real roots' means b² − 4ac ≥ 0. For a line meeting a curve, substitute first, then use the discriminant.

• Leading coefficient a: $a{x}^2+bx+c=a{(x+\frac{b}{2a})}^2+c-\frac{b^2}{4a}$
• Turning point: $y=a{(x+p)}^2+q\Rightarrow (-p,q)$

x

the variable

a

coefficient of x², a ≠ 0

b, c

coefficients of x and the constant

y

the value of the quadratic

p, q

constants in the completed-square form

For ax² + bx + c take out a first. The turning point of y = a(x + p)² + q is (−p, q).

• Factor (ax − b): $\mathrm{f}\left(\frac{b}{a}\right)=0\iff (ax-b)\text{ is a factor}$

f(x)

a polynomial in x

a

a number (for the ax − b form, a ≠ 0)

b

a number

Try factors of the constant term. Mind the sign: f(−2) = 0 gives the factor (x + 2).

f(x)

the polynomial divided

q(x)

the quotient

r

the remainder

a

a number

x

the variable

Not named in every specification, but algebraic division gives the same remainder. Dividing by (ax − b), a ≠ 0, leaves f(b/a).

a, b

any numbers or expressions

Spot it in factorising and cancelling: x² − 9 = (x − 3)(x + 3). A sum of two squares does not factorise.

• Inequality inside: $\left|x\right|<a\iff -a<x<a$
• Inequality outside: $\left|x\right|>a\iff x<-a\text{ or }x>a$

x

a real number

|x| < a means −a < x < a; |x| > a means x < −a or x > a. Solve |f(x)| = g(x) by sketching first.

f, g

functions

x

in the domain of g, with g(x) in the domain of f

fg(x) means do g first. fg and gf are usually different.

• Other order: $\mathrm{f}{\mathrm{f}}^{-1}\left(x\right)=x\text{ for }x\text{ in the range of f}$
• Domain and range: $\text{domain of }{\mathrm{f}}^{-1}=\text{range of f},\text{ range of }{\mathrm{f}}^{-1}=\text{domain of f}$

f

a one-one function

f⁻¹

its inverse

x

an input value

y

the output in y = f(x)

Only one-one functions have inverses. The graph of y = f⁻¹(x) is the reflection of y = f(x) in y = x.

• Vertical: $y=\mathrm{f}\left(x\right)+a\to \text{ translation }\left(\begin{array}{c}0\\ a\end{array}\right)$

a

constant

f

the original function

x, y

coordinates on the graph

Inside the bracket acts on x and goes the opposite way: f(x + 3) moves the graph 3 to the left. Combinations of transformations are A level.

• Vertical stretch: $y=a\mathrm{f}\left(x\right)\to \text{ stretch parallel to the }y\text{-axis, scale factor }a$
• Reflections: $y=-\mathrm{f}\left(x\right),y=\mathrm{f}\left(-x\right)$

a

positive constant, a > 0

f

the original function

x, y

coordinates on the graph

For a > 0, f(ax) uses scale factor 1/a, not a. y = −f(x) reflects in the x-axis and y = f(−x) in the y-axis, so a negative a is a stretch by |a| (or 1/|a|) with a reflection.

• Inverse proportion: $y\propto \frac{1}{x}\iff y=\frac{k}{x}$

k

constant of proportionality

x, y

the two related quantities

Find k from one pair of values, then use the equation.

• Three linear factors: $\frac{p{x}^2+qx+r}{(x-a)(x-b)(x-c)}\equiv \frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}$

A, B, C

constants to find

a, b, c

distinct numbers (the values that make a factor zero)

p, q, r

numerator coefficients (numerator of lower degree than the denominator)

x

the variable

Multiply through, then substitute x = a and x = b. The numerator must have lower degree than the denominator.

A, B, C

constants to find

a, b

numbers with a ≠ b

p, q, r

numerator coefficients

x

the variable

A squared factor needs both A/(x − a) and B/(x − a)². Find A by comparing coefficients.