# 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.

- Clear any fractions and multiply out any brackets. To clear fractions, multiply both sides by a suitable expression.
- 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.
- If more than one term contains the required subject, then take it out as a common factor. This gives an equation of the form $$ \text{an expression} \times \text{the required subject} = \text{an expression} $$
- 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.