Math cheatsheet

Rearranging formulas by clearing algebraic fractions

Carry out a sequence of steps. In each step, do one of the following:

  • do the same thing to both sides
  • simplify one side or both sides
  • swap the sides

Aim to the the following, in order.

  1. Clear any fractions and multiply out any brackets. To clear fractions, multiply both sides by a suitable expression.
  2. Add or subtract terms on both sides to get all the terms containing the required subject on one side, and all the other terms on the other side.
  3. If more than one term contains the required subject, then take it out as a common factor. This gives an equation of the form $$ \text{an expression} \times \text{the required subject} = \text{an expression} $$
  4. Divide both sides by the expression that multiplies the required subject.

Index laws

$$ \begin{aligned} a^m \times a^n = a^{m + n} \qquad \frac{a^m}{a^n} = a^{m-n} \\[1em] (a^m)^n = a^{mn} \\[1em] (a \times b)^n = a^n \times b^n \qquad (\frac{a}{b})^n = \frac{a^n}{b^n} \\[1em] a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \\[1em] a^\frac{1}{n} = \sqrt[n]{a} \qquad a^\frac{m}{n} = (\sqrt[n]{a})^m \end{aligned} $$

Squaring brackets

$$ (x + p)^2 = x^2 + 2px + p^2 $$

$$ (x - p)^2 = x^2 - 2px + p^2 $$

Shapes of $y = ax^2$ parabola

  • u-shaped when the coefficient $a$ is positive
  • n-shaped when the coefficient $a$ is negative

Methods to find the vertex of a parabola from its equation

  • Use the formula $x = -b / (2a)$ to find the $x$-coordinate, then substitute into the equation of the parabola to find the $y$-coordinate
  • Find the $x$-intercepts (if there are any); then the value halfway between them is the $x$-coordinate of the vertex. Find the $y$-coordinate by substituting into the equation of the parabola.
  • Complete the square: the parabola with equation $y = a(x - h)^2 + k$ has vertex $(h, k)$.
  • Plot the parabola and read off the approximate coordinates of the vertex.

The number of solutions of a quadratic equation

The value $b^2 - 4ac$ is called the discriminant of the quadratic expression $ax^2 + bx + c$. Such an equation has:

  • two solutions if $b^2 - 4ac > 0$ (the discriminant is positive).
  • one solution if $b^2 - 4ac = 0$ (the discriminant is zero).
  • no solutions if $b^2 - 4ac < 0$ (the discriminant is negative).

Asymptote of a curve

Is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

Even and odd functions

$f(x) = x^n$ is an even function if $n$ is an even integer and odd if $n$ is an odd integer.