Slope: $\displaystyle\frac{y_1-y_2}{x_1-x_2}$
Slope$_{\perp}$: $\displaystyle m_2 = -\frac{1}{m_1}$
Y-Intercept form: $y=mx+b$
Standard form: $Ax+By=C \mid A,B,C \in \mathbb{Z}$
Point-Slope form: $(y - y_1) = m(x-x_1)$
Transformations using a parent graph: $g(x) = af(bx-h)+k$
Given a point on the parent graph, the point on the transformed graph is $\displaystyle \bigg( \frac{x+h}{b} , ay+k \bigg)$
Vertex form of a parabola: $(y-k) = a(x-h)^2$
Vertex x coordinate of a parabola: $\displaystyle h=\frac{-b}{2a}$
Quadratic Formula: $\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
Standard form for a parabola: $f(x) = ax^2+bx+c$
Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
Perfect Square Trinomial:
Sum/Difference of cubes:
iff $f(x) = x^2$, then $f^{-1}(x) = \pm\sqrt{x}$
iff $\displaystyle f(x) = 4x + 2$, then $\displaystyle f^{-1}(x) = \frac{x}{4} - \frac{1}{2}$
Square root function with transformations: $f(x) = a\sqrt{bx-h} + k$
See above for more information on transformations
Exponential growth: $\displaystyle A(t) = A_0(1+r)^t$
Exponential decay: $\displaystyle A(t) = A_0(1-r)^t$
Exponential growth/decay: $\displaystyle f(x) = a*b^{x-h}+k$
Compound Interest: $\displaystyle A(t) = P\Big(1 + \frac{r}{n}\Big)^{nt}$
Compound Interest (compounding continuously): $\displaystyle A(t) = Pe^{rt}$
Exponential growth with Euler: $\displaystyle f(x) = e^x$
Direct Variation: $y=kx$
Indirect Variation: $\displaystyle y=\frac{k}{x}$
Rational Function: $\displaystyle f(x) = \frac{p(x)}{q(x)}$
$\displaystyle y= \frac{ax+b}{cx+d}$
- HA @ $\displaystyle y=\frac{a}{c}$
- VA: Number that makes the bottom = 0
Midpoint: $\displaystyle \bigg( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} \bigg)$
Distance Formula: $\displaystyle \sqrt{(x_2+x_1)^2 + (y_2+y_1)^2}$
Circle: $\displaystyle (x-h)^2 + (y-k)^2 = r^2$
Ellipse: $\displaystyle \bigg(\frac{x-h}{r_x}\bigg)^2 + \bigg(\frac{y-k}{r_y}\bigg)^2 = 1$
Focal distance for ellipses: $\displaystyle c^2 = big^2 - small^2$
Parabola focal distance: $\displaystyle a = \frac{1}{4p}$
Hyperbola:
$\displaystyle \bigg(\frac{y-k}{r_y}\bigg)^2 - \bigg(\frac{x-h}{r_x}\bigg)^2 = 1$
$\displaystyle \bigg(\frac{x-h}{r_x}\bigg)^2 - \bigg(\frac{y-k}{r_y}\bigg)^2 = 1$
Focal distance: $\displaystyle c^2 = big^2 + small^2$
$\displaystyle _nP_r = \frac{n!}{(n-r)!}$
$\displaystyle _nC_r = \frac{n!}{r!(n-r)!}$
Probability of event happening: $\displaystyle P(E) = \frac{n(E)}{n(S)}$
Probability: $\displaystyle P(\text{A or B}) = P(A) + P(B) - P(\text{A and B})$
Odds: $\displaystyle \frac{\text{number of possibilties where event happens}}{\text{number of possibilities where event doesn't happen}}$
Probability of a complement: $\displaystyle P(A^c) = 1 - P(A)$
Population standard deviation $\displaystyle \sigma = \sqrt{\frac{(x_1-\overline{x})^2 + (x_2-\overline{x})^2 + \cdots + (x_n-\overline{x})^2}{n} }$
Sample standard deviation $\displaystyle S_x = \sqrt{\frac{(x_1-\overline{x})^2 + (x_2-\overline{x})^2 + \cdots + (x_n-\overline{x})^2}{n-1} }$
Population Z-score: $\displaystyle z = \frac{x-\mu}{\sigma}$
Sample Z-score: $\displaystyle z = \frac{x-\overline{x}}{S_x}$
nth term of an arithmetic sequence: $\displaystyle t(n) = t(1) + (n-1)d$
nth term of a geometric sequence: $\displaystyle t(n) = t(1) * r^{n-1}$
Partial sum of an arithmetic series: $\displaystyle \sum_{i=1}^{n} t(i) = \bigg(\frac{n}{2}\bigg)(t_1+t_n) $
Partial sum of a geometric series: $\displaystyle \sum_{i=1}^n t(i) = t_1\bigg(\frac{1-r^n}{1-r}\bigg)$
if $0 < r < 1$ then the infinite sum exists: $\displaystyle \sum_{i=1}^\infty t(i) = \frac{t_1}{1-r}$