act math

vocabulary

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. 2\sqrt2 or π\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

real number: any number that can be found on a number line

prime number: a number greater than 1 that has no factors besides 1 and itself

geometry

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=4sP=4s ; A=s2A=s^2

rectangle: P=2l+2wP=2l+2w ; A=lwA=lw

triangle: P = sum of all sides ; A=12bhA=\frac12bh

circle: C=2πrC=2\pi r ; A=πr2A=\pi r^2

midpoint: m=x1+x22,y1+y22m=\frac{x1+x2}{2},\frac{y1+y2}{2}

distances on a coordinate plane: d=(x2−x1)2+(y2−y1)2d=\sqrt{\left(x2-x1\right)^2+\left(y2-y1\right)^2}

pythagorean theorem: a2+b2=c2a^2+b^2=c^2

slope of a line: m=y2−y1x2−x1m=\frac{y2-y1}{x2-x1}

equation of a line (slope intercept form): y=mx+by=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 - −18=8-\frac18=8

multiply binomials using FOIL: first, outer, inner, last

trigonometric ratios (SOHCAHTOA): sin⁡=oppositehypotenuse,cos⁡=adjacenthypotenuse,tan⁡=oppositeadjacent\sin=\frac{opposite}{hypotenuse},\cos=\frac{adjacent}{hypotenuse},\tan=\frac{opposite}{adjacent}

point-slope form: (y−y1)=m(x−x1)\left(y-y1\right)=m\left(x-x1\right)

rectangular solid: V=lwhV=lwh

right cylinder: V=πr2hV=\pi r^2h

algebra

direct substitution: 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…dt=rtt…dt=rd=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; xy=x⋅y\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: an=a1+(n+1)da_{n}=a_1+\left(n+1\right)d

geometric sequences: an=a1⋅r(n−1)a_{n}=a_1\cdot r^{\left(n-1\right)}

difference of squares:a2−b2a^2-b^2

complex numbers:

i0,i4=1i^0,i^4=1

i1,i5=ii^1,i^5=i

i2,i6=−1i^2,i^6=-1

i3,i7=−ii^3,i^7=-i

to find the nth term of a repeating decimal: given digit number / number of digits in repeating decimal

adding subtracting fractions: ensure that the fractions have the same denominator; if they don’t …

  1. find their least common multiple (lcm) ; ex: lcm of 4 and 6 is 12

  2. after finding the lcm, multiply the original numbers by a whole fraction to covert them

  3. add / subtract

multiplying fractions: simply multiply across numerators and denominators

dividing fractions: keep change flip, multiply across numerators and denominators

statistics and probability

using percents: “what number/percent” (n/p), “is” (=), “of” (*); ex - n=1.4⋅60;75=p(215)n=1.4\cdot60;75=p\left(215\right)

average speed: disttime\frac{dist}{time}

sum: avg⋅#termsavg\cdot\#terms

probability: desiredtotal\frac{desired}{total}

percent change: (new−given)given⋅100\frac{\left(new-given\right)}{given}\cdot100

increasing percent value:

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

  2. multiply this number by the other total number

decreasing percent value:

  1. subtract given percent from 100

  2. multiply this number by other total number