# PreCalculus 12

$$(x - a)(x - b)(x - c) = 0 \implies x \in \{a, b, c\}$$

$$P(k) = 0 \implies (x - k) \; \text{is a factor of} \; P(x)$$

$$y = \frac{P_m(x)}{Q_n(x)} = \frac{a_m x^m + \dots + a_0}{b_n x^n + \dots + b_0}$$

when horizontal asymptote oblique asymptote
m < n y = 0 no
m = n $$y = \frac{a_m}{b_n}$$ no
m = n + 1 no $$y = P_m(x) \div Q_n(x)$$
m > n + 1 no no

original transformed
function $$y = f(x)$$ $$y = s \, f\Big(c \, (x - h)\Big) + v$$
points $$\Big( x, \: y \Big)$$ $$\Big( \frac{x}{c} + h, \: s \cdot y + v \Big)$$
parameter effect
s stretch vertically (the y-axis)
c compress horizontally (the x-axis)
h move left/right (“backwards” horizontal movement)
v move up/down (vertical movement)

$$b^x \cdot b^y = b^{x + y}$$

$$\frac{b^x}{b^y} = b^{x - y}$$

$$b^{x^y} = b^{xy}$$

$$\sqrt[n]{b^m} = b^\frac{m}{n}$$

$$b^0 = 1$$

$$log_b(1) = 0$$

$$x = b^{log_b(x)}$$

$$log_b(xy) = log_b(x) + log_b(y)$$

$$log_b(x^y) = y \cdot log_b(x)$$

$$log_b(x) = \frac{log_k(x)}{log_k(b)}$$

 $$tan(x) = \frac{sin(x)}{cos(x)}$$ $$cot(x) = \frac{cos(x)}{sin(x)}$$ $$csc(x) = \frac{1}{sin(x)}$$ $$sec(x) = \frac{1}{cos(x)}$$ $$cot(x) = \frac{sec(x)}{csc(x)}$$ $$sin^2(x) + cos^2(x) = 1$$ $$cot^2(x) + 1 = csc^2(x)$$ $$tan^2(x) + 1 = sec^2(x)$$