Formula sheet · Grades 10 to 12

Precalculus formulas

12 formulas from this course. Each one gives the rule in math, one line on what it means, and a worked example that uses it.

Domain rules

\text{denominator} \neq 0,\qquad \text{radicand} \geq 0

Exclude inputs that divide by zero or take the square root of a negative number.

f(x) \Rightarrow \text{replace every } x \text{ with the input}

Substitute the input into the rule and simplify with the order of operations.

f(x) = mx + b \Rightarrow f^{-1}(x) = \frac{x - b}{m}

Swap the input and the output and solve for the new output to undo the function.

\log_b x = y \Leftrightarrow b^y = x

A logarithm is the exponent that turns the base into the argument.

\log_b(MN) = \log_b M + \log_b N

A product inside a logarithm splits into the sum of two logarithms.

\log_b(M^p) = p\log_b M

An exponent inside a logarithm moves to the front as a multiplier.

b^m = b^n \Rightarrow m = n

Write both sides as powers of the same base, then set the exponents equal.

a_n = a_1 + (n - 1)d

Each term adds the common difference d to the first term.

a_n = a_1 r^{n-1}

Each term multiplies the first term by the common ratio r, one fewer time than its position.

S_n = \frac{n}{2}\left(2a_1 + (n - 1)d\right)

The total of the first n terms of an arithmetic sequence.

S_n = \frac{a(1 - r^n)}{1 - r}

The total of the first n terms when each term multiplies by r.

S = \frac{a}{1 - r},\qquad |r| < 1

When the ratio is between -1 and 1, the infinite series adds up to a finite total.

Practice problems with answers for every Precalculus topic.