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)

Formulas

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.