Basics

• Fraction: $\text{fraction} = \frac{\text{part}}{\text{whole}}$
• Percent: $\text{percent} = \frac{\text{part}}{100}$
• Percent change: $\frac{|\text{old}-\text{new}|}{\text{old}} \times 100\%$

Percent Operations

• Increase by $p\%$: $\times\left(1+\frac{p}{100}\right)$
• Decrease by $p\%$: $\times\left(1-\frac{p}{100}\right)$
• Multiplier: $1 \pm \frac{p}{100}$

Interest Formulas

• Simple: $A = P(1+rt)$
• Compound annually: $A = P(1+r)^t$
• Compound $n$ times/year: $A = P\left(1+\frac{r}{n}\right)^{nt}$

Exponential Model

• $f(t) = A \cdot b^{t/p}$
  $A$ = starting amount, $b$ = factor per period, $p$ = period length

Linear Units

• $10\text{ft} \times \frac{12\text{in}}{1\text{ft}} = 120\text{in}$

Square Units

• $10\text{ft}^2 \times \left(\frac{12\text{in}}{1\text{ft}}\right)^2 = 1440\text{in}^2$
  Square the conversion factor for area!

Volume Units

• $10\text{ft}^3 \times \left(\frac{12\text{in}}{1\text{ft}}\right)^3 = 17280\text{in}^3$
  Cube the conversion factor for volume!

Standard Form

• $f(x) = ax^2 + bx + c$
• Vertex: $\left(-\frac{b}{2a},\ f\left(-\frac{b}{2a}\right)\right)$
• Vertex $x$: $x = -\frac{b}{2a}$
• $y$-intercept: $c$
• $x$-intercepts: $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Other Forms

• Factored: $f(x) = a(x-m)(x-n)$
• Vertex form: $f(x) = a(x-h)^2 + k$

Vieta's Formulas

• Sum of roots: $-\frac{b}{a}$
• Product of roots: $\frac{c}{a}$

Discriminant

• $D = b^2 - 4ac$
• Positive: 2 real solutions
• Zero: 1 real solution
• Negative: 0 real solutions

Slope Formula

• $m = \frac{y_2-y_1}{x_2-x_1}$
• Horizontal line: $m = 0$
• Vertical line: undefined

Line Forms

• Standard: $Ax + By = C$
• $x$-intercept: $\frac{C}{A}$
• Slope-intercept: $y = mx + b$
• Point-slope: $y - y_1 = m(x - x_1)$

Distance & Midpoint

• $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
  Desmos: midpoint() and distance()

Parallel & Perpendicular

• Parallel: equal slopes
• Perpendicular slopes: $m_1 m_2 = -1$

Exponent Rules

• $a^b \cdot a^c = a^{b+c}$
• $\frac{a^b}{a^c} = a^{b-c}$
• $(a^b)^c = a^{bc}$
• $a^{-b} = \frac{1}{a^b}$

Fractional Exponents

• $a^{b/c} = \sqrt[c]{a^b} = \left(\sqrt[c]{a}\right)^b$

For $h(t) = at^2 + bt + c$:

• Initial height: $h(0) = c$
• Time at max: $t = -\frac{b}{2a}$
• Max height: $h\left(-\frac{b}{2a}\right)$

Launch (at $t = 0$)

• Height $= c$ (the $y$-intercept)

Landing (positive $x$-intercept)

• Solve $h(t) = 0$
  Use the positive solution when time is the context

SOH-CAH-TOA

right triangles only!

• $\sin = \frac{\text{opp}}{\text{hyp}}$
• $\cos = \frac{\text{adj}}{\text{hyp}}$
• $\tan = \frac{\text{opp}}{\text{adj}}$

Right Triangles

• $a^2 + b^2 = c^2$
• Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Special Right Triangles

• $45°\text{-}45°\text{-}90°$: $x : x : x\sqrt{2}$
• $30°\text{-}60°\text{-}90°$: $x : x\sqrt{3} : 2x$

Complementary Angles

• $\sin(x) = \cos(90° - x)$

Conversion

• $360° = 2\pi \text{ radians}$

Statistics

• Mean: $\text{mean} = \frac{\text{sum of items}}{\text{number of items}}$
• Weighted mean: $\frac{\sum x_i n_i}{\sum n_i}$
• Median: middle number after sorting
• Range: $\text{range} = \max - \min$

Probability

• $P = \frac{\text{favorable outcomes}}{\text{total possible outcomes}}$
• $P(A \mid B) = \frac{\text{outcomes in both } A \text{ and } B}{\text{outcomes in } B}$
• Complement: $P(\text{not } A) = 1 - P(A)$
  Desmos tip: Use mean(1,2,3,4,5) and median(1,2,3,4,5)

For $Ax + By = C$:

• Slope: $-\frac{A}{B}$
• $y$-intercept: $\frac{C}{B}$

Number of Solutions

For two standard-form lines:

• $A_1x + B_1y = C_1$
• $A_2x + B_2y = C_2$
• One solution: $\frac{A_1}{A_2} \neq \frac{B_1}{B_2}$
• Infinite solutions: $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$
• No solutions: $\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}$

Angle Facts

• Vertical angles are congruent
• Linear pair: add to $180°$
• Circle: add to $360°$

Parallel Lines + Transversal

• All acute angles are congruent
• All obtuse angles are congruent

Triangles

• Angles sum to $180°$
• Exterior angle = sum of remote interior angles
• Similar figures: matching side lengths scale by $k$
• Area scale: $k^2$

Circles

• Radius $\perp$ tangent line
• Central angle $= 2\times$ inscribed angle
• A diameter subtends a right angle

Circle Equation

• $(x-h)^2 + (y-k)^2 = r^2$
• Center: $(h, k)$
• Radius: $r$

Expanded Circle

$x^2 + y^2 + Ax + By + C = 0$

• Center: $\left(-\frac{A}{2}, -\frac{B}{2}\right)$
• $r^2$: $\left(\frac{A}{2}\right)^2 + \left(\frac{B}{2}\right)^2 - C$

Arcs & Sectors

• Circumference: $C = 2\pi r = \pi d$
• Area: $A = \pi r^2$
• Arc length: $s = \frac{\theta}{360°} \cdot 2\pi r$
• Sector area: $A = \frac{\theta}{360°} \cdot \pi r^2$

Factoring Patterns

• Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
• Perfect square: $a^2 + 2ab + b^2 = (a+b)^2$
• Perfect square: $a^2 - 2ab + b^2 = (a-b)^2$

Completing the Square

• $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

Areas

• Semicircle area: $A = \frac{1}{2}\pi r^2$

Volumes

• Rectangular prism: $V = lwh$
• Cube: $V = s^3$
• Cylinder: $V = \pi r^2 h$
• Cone: $V = \frac{1}{3}\pi r^2 h$
• Sphere: $V = \frac{4}{3}\pi r^3$
• Hemisphere: $V = \frac{2}{3}\pi r^3$

Surface Area

• Rectangular prism: $SA = 2(lw + lh + wh)$
• Cube: $SA = 6s^2$
• Cylinder: $SA = 2\pi r^2 + 2\pi rh$
• Sphere: $SA = 4\pi r^2$

Scaling

• Lengths: $\times k$
• Areas: $\times k^2$
• Volumes: $\times k^3$

Transformations

• $f(x) + k$: shifts up/down by $k$
• $f(x-h)$: shifts left/right by $h$
• $f(-x)$: reflects over the $y$-axis
• $-f(x)$: reflects over the $x$-axis
• $a \cdot f(x)$: vertical stretch/compress by $a$
• $f(bx)$: horizontal stretch/compress by $\frac{1}{b}$

Composition

• $f(g(x))$: apply $g$ first, then $f$

Definition

• $|x| = x$ if $x \geq 0$
• $|x| = -x$ if $x < 0$

Solving Equations

• $|x| = a \Rightarrow x = a$ or $x = -a$ (when $a \geq 0$)
• $|x| = -a$ when $a < 0$

Graphing

• V-shape; vertex at $(h, k)$ for $y = |x-h| + k$

Inequalities

• $|x| < a \Rightarrow -a < x < a$
• $|x| > a \Rightarrow x < -a$ or $x > a$

Simplifying

• Factor numerator & denominator, cancel common factors

Adding / Subtracting

• Rewrite with a common denominator first

Excluded Values

• Denominator $\neq 0$ — exclude those $x$ values

Arithmetic

• $a_n = a_1 + (n-1)d$, where $d$ = common difference

Geometric

• $a_n = a_1 \cdot r^{n-1}$, where $r$ = common ratio

Remainder Theorem

• $f(a)$ = remainder when dividing $f(x)$ by $(x-a)$

Factor Theorem

• $(x-a)$ is a factor of $f(x)$ iff $f(a) = 0$

Zero Product Property

• If $ab = 0$, then $a = 0$ or $b = 0$

Basics

• $i = \sqrt{-1}$
• $i^2 = -1$

Operations

• $(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i$
• $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Compound Inequalities

• $a < x < b$ means both $a < x$ and $x < b$

Flipping the Sign

• Multiplying/dividing by a negative flips the inequality direction

Unit Rate

• $\text{rate} = \frac{\text{quantity}}{\text{unit}}$

Dimensional Analysis

• Multiply by conversion factors to cancel units

Variation

• Direct: $y = kx$
• Inverse: $y = \frac{k}{x}$