# Properties

Mon 01 January 2018

## Order of operations

1. Exponents/Roots
2. Products/Division
3. Add / Subtract

## Properties

1. Associative property: a + b + c = (a + b) + c = a + (b + c) and abc = (ab)c = a(bc)
2. Commutative property: a + b = b + a and ab = ba
3. Distributive property: a(b + c) = ab + ac

Addition commutative, which means that a + b = b + a. It is also associative, which means that if you have a long summation like a + b + c you can compute it in any order (a + b) + c or a + (c + b)

## Common functions and their inverses

$$\frac{1}{x^{2}}$$ is the same as $$x^{-2}$$

\begin{aligned} x+2 &= 2+x \\ 2^x &= \log_2(x) \end{aligned}
$$3x + 5 ⇔ \frac{1}{3}(x - 5)$$

Returns e^x, the inverse function of \log(x).

$$a^x ⇔ log_a (x)$$
$$exp(x) ≡ e x ⇔ ln(x) ≡ log_e(x)$$
$$sin(x) ⇔ sin−1 (x) ≡ arcsin(x)$$
$$cos(x) ⇔ cos−1 (x) ≡ arccos(x)$$

The function-inverse relationship is reflexive—

###### Completing the square

Any quadratic expression $$Ax^2 + Bx + C$$ can be rewritten in the form $$A(x − h)^2 + k$$ for some constants h and k. Geometrically h shifts the graph right and k up.