PreCalculus 11
$$(x - a)(x - b)(x - c) = 0 \implies x \in \{a, b, c\}$$
$$P(k) = 0 \implies (x - k) \; \text{is a factor of} \; P(x)$$
$$y = \frac{P_m(x)}{Q_n(x)} = \frac{a_m x^m + \dots + a_0}{b_n x^n + \dots + b_0}$$
| 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) |
$$b^x \cdot b^y = b^{x + y}$$
$$\frac{b^x}{b^y} = b^{x - y}$$
$$b^{x^y} = b^{xy}$$
$$\sqrt[n]{b^m} = b^\frac{m}{n}$$
$$b^0 = 1$$
$$log_b(1) = 0$$
$$x = b^{log_b(x)}$$
$$log_b(xy) = log_b(x) + log_b(y)$$
$$log_b(x^y) = y \cdot log_b(x)$$
$$log_b(x) = \frac{log_k(x)}{log_k(b)}$$
| $$tan(x) = \frac{sin(x)}{cos(x)}$$ | $$cot(x) = \frac{cos(x)}{sin(x)}$$ | |
| $$csc(x) = \frac{1}{sin(x)}$$ | $$sec(x) = \frac{1}{cos(x)}$$ | $$cot(x) = \frac{sec(x)}{csc(x)}$$ |
| $$sin^2(x) + cos^2(x) = 1$$ | $$cot^2(x) + 1 = csc^2(x)$$ | $$tan^2(x) + 1 = sec^2(x)$$ |
