To find $23\%$ of $70$, take $.23 \times 70$. $23$ is what percent of $70$? $\dfrac{23}{70} \times 100$.
$23$ is $70\%$ of what? $23 = .7x, \dfrac{23}{.7}=x$.
A $10\%$ increase followed by a $20\%$ increase is $(1.1\times 1.2) – 1\times 100 = 32\%$.
$\text{Distance} = \text{Rate} \times \text{time}$
Midpoint formula:$\Bigg(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\Bigg)$
Slope formula: $\dfrac{y_1 – y_2}{x_1 – x_2}$
Distance formula: $\sqrt{(x_1 – x_2)^2 + (y_1 – y_2)^2 }$
If two lines are perpendicular, their slopes are negative reciprocals.
Given a point $(x_1, y_1)$ and a slope $m$, the equation of the line is $y – y_1 = m(x – x_1)$
Alternatively, you can use $y_1 = mx_1+ b$, and solve for $b$
Slope of line in form $ax + by = c$ is $-\dfrac{a}{b}$
In slope intercept form: $by = -ax + c \rightarrow y = -\dfrac{a}{b}x+\dfrac{c}{b}$
Triangle inequality: if two sides have length $a$ and $b$, the other side $c$ is $|a – b| < c < a + b$
A $30-60-90$ triangle has side lengths with ratios $1-\sqrt{3}-2$
A $45-45-90$ triangle has side lengths with ratios $1-1-\sqrt{2}$
Volume of a box: $lwh$
Volume of a cube: $s^3$
Surface area of a box: $2(lw+lh+wh)$
Surface area of a cube: $6s^2$
Diagonal of a box: $\sqrt{l^2+w^2+h^2}$
Diagonal of a cube: $\sqrt{3}s^3$
If a cube is inscribed in a sphere, the diagonal of the cube = the diameter of the sphere.
Volume of a cone: $\dfrac{\pi r^2 h}{3}$
Surface area of a cylinder: $2 \pi r^2 + 2\pi r h$
Surface area of a sphere: $4\pi r^2$
Volume of a sphere: $\dfrac{4\pi r^3}{3}$
Area of an equilateral triangle: $\dfrac{\sqrt{3}s^2}{4}$
Area of a parallelogram: $bh$
Area of a trapezoid: $\dfrac{(b_1 + b_2)h}{2}$
Rhombus area formula: $\dfrac{d_1 d_2}{2}$
Perimeter of a rectangle: $2l+2w$
Circumference of circle: $2\pi r$
Area of a circle: $\pi r^2$
Volume of cylinder: $\pi r^2 h$
Surface area of cylinder: $2 \pi r^2 + 2 \pi r h$
The diagonals of a rhombus are perpendicular.
Sum of interior angles of polygon: $180^{\circ}(n-2)$
The sum of the roots of a quadratic: $-\dfrac{b}{a}$.
The quadratic formula: $\dfrac{-b \pm \sqrt{b^2 – 4ac} }{2a}$
$x$-coordinate of the vertex of a parabola: $-\dfrac{b}{2a}$
$y = (x – a)^2 + b$ is a parabola in vertex form. Its vertex is $(a,b)$.
To find the equation of a quadratic given $x$-intercepts $a$ and $b$, take $y = (x-a)(x-b)$ and FOIL it out.
Sum and difference of cubes formulas: $a^3+b^3 =(a+b)(a^2-ab+b^2)$, $a^3-b^3 =(a-b)(a^2+ab+b^2)$
$\text{cot} = \dfrac{1}{\text{tan}} \quad \text{csc} = \dfrac{1}{\text{sin}} \quad \text{sec} = \dfrac{1}{\text{cos}}$
$\text{sin} = \dfrac{\text{opp}}{\text{hyp}} \quad \text{cos} = \dfrac{\text{adj}}{\text{hyp}} \quad \text{tan} = \dfrac{\text{opp}}{\text{adj}}$
Radians to degrees: $\dfrac{\pi}{180^{\circ}}$
Degrees to radians: $\dfrac{180^{\circ}}{\pi}$
Law of sines: $\dfrac{\text{sin } A}{a} = \dfrac{\text{sin } B}{b} = \dfrac{\text{sin } C}{c}$
Law of cosines: $c^2 = a^2 + b^2 – 2ab \text{ cos } C$ or $\text{cos } C = \dfrac{a^2 + b^2 + c^2}{2ab}$
$\text{sin }^2 x + \text{cos}^2 x = 1, \quad \text{sin}^2 x = 1 – \text{cos}^2 x, \quad \text{cos}^2 x = 1 – \text{sin}^2 x$
$\text{tan }^2 x + 1 = \text{sec}^2 x, \quad 1 + \text{csc}^2 x = \text{cot}^2 x, \quad \text{sec}^2 x – \text{tan}^2 x = 1$
$\text{sin } 2x = 2 \text{sin } x \text{cos } x, \quad \text{cos } 2x = \text{cos}^2 x – \text{sin}^2 x = 2 \text{cos}^2 x – 1 = 1 – 2\text{sin}^2 x$
$\text{sin } 30^{\circ} = \text{cos } 60^{\circ} = \dfrac{1}{2}, \text{sin } 60^{\circ} = \text{cos } 30^{\circ} = \dfrac{\sqrt{3}}{2},$ and
$\text{sin } 45^{\circ} = \text{cos } 45^{\circ} = \dfrac{\sqrt{2}}{2}, \text{tan } 30^{\circ} = \dfrac{\sqrt{3}}{3},$ and $\text{tan } 60^{\circ} = \sqrt{3}$
The trigonometric area formula is $\text{Area} = \dfrac{ab \text{ sin } A}{2}$
In the function $y = a \text{ sin } bx, |a|$ is the amplitude and $\dfrac{2\pi}{b}$ is the period.
$\text{log } ab = \text{log } a + \text{log } b$
$\text{log } \dfrac{a}{b} = \text{log } a – \text{log } b$
$\text{log } a^b = b \text{log } a$
$\text{arc length} = 2 \pi r \dfrac{\theta}{360}$
$\text{area of pie slice} = \pi r^2 \dfrac{\theta}{360}$
Circle with center $(x_1,y_1)$ and radius $r$ is $(x-x_1)+(y-y_1)^2=r^2$
Factorial: $n! = n(n-1)(n-2) \cdots 1$
Combinations: pick $k$ of $n$: $\dfrac{n!}{k!(n-k)!}$
Permutations: $\dfrac{n!}{(n-k)!}$
The number of permutations of the letters in MISSISSIPPI is $\dfrac{11!}{(4! \times 4! \times 2!)}$
A sum of arithmetic series formula is $a_1 + a_2 + \cdots + a_n$ is $\dfrac{(a_1 + a_n) \times n}{2}$
The sum of an infinite geometric series: $a_1 + a_2 + \cdots = \dfrac{a_1}{(1-r)}$
for depreciation, $A = P (1 – r)^t$
for simple interest, use $A = P(1+r)^t$
for compound interest, use $A = P \Bigg( 1 + \dfrac{r}{n} \Bigg)^{nt}$
for continuous interest, use $A= Pe^{rt}$
Contrapositive: “All tables are flat” is the same as “If it is not a table, it is not flat”. Switch them and make them both not, and it is the same thing.
Rotating $90^{\circ}$ clockwise, $(a,b) \rightarrow (-b,a)$,
Rotating $90^{\circ}$ counterclockwise, $(a,b) \rightarrow (b,-a)$
Rotating $180^{\circ} : (a,b) \rightarrow (-b,-a)$
Reflecting about the $x$-axis, $(a,b) \rightarrow (a,-b)$
Reflecting about the $y$-axis, $(a,b) \rightarrow (-a,b)$
If $y = x^2$ is moved $3$ up and $5$ to the right, the new equation is $y =(x-5)^2+3$.
Similarly, if the circle $x^2 + y^2 = 36$ is moved $3$ up and $5$ to the right, the new equation is $(x-5)^2 +(y-3)^2 = 36$.
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