act math

integer

a whole number (positive, negative, or zero)

rational number

any number that can be written as a fraction (includes terminating and repeating decimals)

irrational number

decimals that never end or repeat (ex. $\sqrt2$ or $\pi$ )

factor

a number that divides another number evenly

multiple

the result of multiplying a number by an integer

median

the middle value in an ordered set

mode

the value that appears most often

range

difference between the highest and lowest value in a set

intercept

the point where a line crosses the x or y axis

coefficient

the numerical factor in front of a variable in an expression

vertex

the turning point of a parabola, or a corner of a geometric figure

congruent

figures that have the same shape and size

similar

figures with the same shape but different yet proportional size

amplitude

half the distance between the maximum and minimum values of a sine function graph

general geometry topics

triangle int. angle sum = 180; comp. angles add up to 90; supp. angles/lines = 180; vertical angles are equal

parallel lines cut by a transversal

lines m and n are parallel; line t cuts through them

corresponding angles

two angles in the same relative position on one side of the transversal; equal

interior angles

the angles between the two parallel lines and split by the transversal; equal

square

$P=4s$ ; $A=s^2$

rectangle

$P=2l+2w$ ; $A=lw$

triangle

P = sum of all sides ; $A=\frac12bh$

circle

$C=2\pi r$ ; $A=\pi r^2$

midpoint

$$m=\frac{x1+x2}{2},\frac{y1+y2}{2}$$

distance on a coordinate plane

$$d=\sqrt{\left(x2-x1\right)^2+\left(y2-y1\right)^2}$$

pythagorean theorem:

$$a^2+b^2=c^2$$

slope of a line:

$$m=\frac{y2-y1}{x2-x1}$$

equation of a line (slope intercept form)

$$y=mx+b$$

slopes of parallel lines

get y by itself > divide everything by y’s coefficient > solve like normal

slopes of perpendicular lines

opposite reciprocals; ex - $-\frac18=8$

multiply binomials using FOIL

$$x1\cdot x2\longrightarrow{}x1\cdot y1\longrightarrow{}x2\cdot y1\longrightarrow{}x2\cdot y2$$

trigonometric ratios (SOHCAHTOA)

$$\sin=\frac{opposite}{hypotenuse},\cos=\frac{adjacent}{hypotenuse},\tan=\frac{opposite}{adjacent}$$

point-slope form

$$\left(y-y1\right)=m\left(x-x1\right)$$

rectangular solid

$$V=lwh$$

right cylinder

$$V=\pi r^2h$$

direct substitution (best for systems of equations)

when the variable’s value is given and needs to be plugged into the equation

solve an equation for a different variable

ex - solve to get r by itself: $d=rt\ldots\frac{d}{t}=\frac{rt}{t}\ldots\frac{d}{t}=r$

exponents

multiply like bases, add the exponents; divide like bases, subtract the exponents; $\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}$

add/subtract polynomials by combining like terms

combining terms that have the same exponents

distributive property

multiply the terms inside the parentheses by the terms outside the parentheses.

absolute value

menu > equation > F3 > OPTN + F6 + F4 + F1 for absolute value > SHIFT + A, O, X for x > SHIFT + . for = > EXE to solve

arithmetic sequences

$$a_{n}=a_1+\left(n+1\right)d$$

geometric sequences

$$a_{n}=a_1\cdot r^{\left(n-1\right)}$$

difference of squares

$a^2-b^2$

complex numbers

$$i^0,i^4=1$$

$$i^1,i^5=i$$

$$i^2,i^6=-1$$

$$i^3,i^7=-i$$

using percents

“what number/percent” (n/p), “is” (=), “of” (*); ex - $n=1.4\cdot60;75=p\left(215\right)$

average speed

$$\frac{dist}{time}$$

sum

$$avg\cdot\#terms$$

probability

$$\frac{desired}{total}$$

percent change

$$\frac{\left(new-given\right)}{given}\cdot100$$

law of cosines

c² = a² + b² - 2abcos(C)

real numbers

any value that can be found on a number line (rational/irrational numbers, integers)

prime numbers

numbers excluding 1 that have no other factors that itself and 1

to find the nth term of a repeating decimal …

given nth term / number of digits in repeating decimal

adding / subtracting decimals

1. find the lcm of the denominators

2. convert by multiplying the original numbers by their respective whole fraction

3. add / subtract

multiply fractions

multiply across numerators and denominators

dividing fractions

1. keep, change, flip

2. multiply across numerators and denominators

increasing percent value

1. increase given percent by 100 (ex. 20% increase = 120%)

2. multiply with other given number

decreasing percent value

1. subtract given percent from 100

2. multiply with other given number

multiplying two 2 × 2 matrices

ae + bg , af + bh

ce + dg , cf + dh

multiplying a 2 × 3 matrix by a 3 × 4 matrix

ag + bh + ci , aj + bk + cl , am + bn + co , ap + bq + cr

dg + eh + fi , dj + ek + fl , dm + en + fo , dp + eq + fr

the fundemental counting principle (UNIQUE, INDEPENDENT OUTCOMES)

multiply the number of options

the fundemental counting principle (DIGIT CODES w/ NUMBERS 0-9, NO RESTRICTIONS APPLIED)

multiply 10 per one digit (ex. 3 digit code = 10 × 10 × 10)

the fundemental counting principle (DIGIT CODES w/ NUMBERS 0-9, RESTRICTIONS APPLIED)

multiply by decreasing order from 10 per one digit (3 digits = 10 × 9 × 8)

solving inequalities for x

solve like a regular equation

be careful multiplying and dividing in equality problems, because …

changing the sign of the number changes the direction of the inequality symbol

reverse FOIL

after factoring, set the factors equal to 0 (this essentially just changes the sign of the numbers)

the quadratic formula

x =−b ± SQ. RT. b² − 4ac / 2a

the discriminant formula

b² - 4ac

vertex form

y = a(x - h)² + k

systems of equations (NO SOLUTION)

lines would be PARALLEL

systems of equation (ALL REAL / INFINITE SOLUTIONS)

lines are EQUIVALENT

when looking at systems of equations …

find the point of intersection, if any

function

y = f(x)

function domain

all possible values of x

function range

all possible values of y

composite functions f(g(x))

to solve for f(g(x)),plug the operations of g(x) into those of f(x).

f(x)3x + 2 , g(x) = 2x -3

f(g(x)) = 3(2x - 3) + 2 = 6x -7.

inverse function

f(x) = 3x + 2

f ⁻¹(x) —> x = 3y + 2

f ⁻¹(x) = x - 2 / 3

translations up and down

add or subtract from y value

addition goes up

subtraction goes down

translations left and right

add or subtract from x value

addition goes left

subtraction goes right

function stretches and compressions

multiplied by constants

stretches are greater than 1

compressions are less than 1

function flip (HORIZONTAL)

multiply the FUNCTION by -1

function flip (VERTICAL)

multiply the X VALUE by -1

ratio

exemplifies a relationship of multiple things by relative size; can be expanded or reduced

combining ratios

1. multiply numbers of the same variable

2. multiply numbers to their corresponding places