ACT Prep Math

quadratic formula

[-b ± √(b² - 4ac)] / (2a)

slope formula

(y2- y1) / (x2- x1)

rationals

fractions (anything expressible as a ratio of integers)

order of operations

PEMDAS (Parentheses (grouping symbols - inside to outside) / Exponents / Multiply & Divide (Left to right) / Add & Subtract(Left to right))

Arithmetic Sequences

t1, t1+ d, t1+2d, ...

Geometric Sequences

t1, t1· r, t1· r², ...

percentages

part = (percent / 100) x whole; is/of=%/100

average or mean

sum of terms / number of terms

average speed

total distance / total time

sum

average x (number of terms)

mode

value in the list that appears most often

median

middle value in the list after values are put in numerical order

fundamental counting principle

if an event can happen in N ways, and another, independent event can happen in M ways, then both events together can happen in N x M ways.

probability

number of desired outcomes / number of total outcomes

adding exponents

x^a · x^b = x^(a+b)

subtracting exponents

x^a / x^b = x^(a-b)

negative exponents

1 / x^b = x^(-b)

multiplying exponents

(x^a)^b = x^(a·b)

x^0

1

roots

√(xy) = √x · √y

FOIL

(x+a)(x+b) = x² + (b+a)x + ab

difference of squares

a² -b² = (a+b)(a-b)

a² + 2ab + b²

(a+b)²

a² - 2ab + b²

(a-b)(a-b)

distance formula

√[(x2- x1)² + (y2- y1)²)]

midpoint formula

[(x1+ x2)/2 , (y1+ y2)/2]

intersecting lines

opposite angles are equal (each pair of angles along the same line add to 180 degrees)

area of triangle

area = (1/2)bh

angles of triangle

add up to 180 degrees

length of triangle sides

length of 1 side is always less than the sum of the other 2

equilateral

three equal sides, all angles = 60 degrees

isosceles

2 equal sides, 2 equal angles

similar

same shape, corresponding angles are equal, corresponding sides are in proportion

full circle

0

length of arc

r * angle (in radians); s=r * theta

rectangle area

length * width

parallelogram area

base * height

rectangular solid volume

length * width * height

right cylinder volume

pi · r² · h

equation of circle

(x - h)² + (y - k)² = r² ; (h,k) is the center

logarithms

y = log(little b)x means b^y = x

complex numbers

a + bi

special sin cos relationship

sin^2(angle) + cos^2(angle) = 1

trapezoid area

[(base1 + base2) / 2] · h

direct variation

y=kx

inverse variation

y=k/x

slope-intercept form

y=mx+b

point-slope form

y - y1= m(x - x1)

standard form

Ax + By=C, where A, B, and C are not decimals or fractions, where A and B are not both zero, and where A is not a negative

Dividing by a negative number in an inequality

must reverse the inequality symbol

Graphing < or > on a coordinate plane

dotted line

Graphing ≥ or ≤ on a coordinate plane

solid line

Graphing ≥ or > on a coordinate plane

shade upwards or to the right

Graphing ≤ or < on a coordinate plane

shade downwards or to the left

chord

The distance from one point on the circle to another point on the circle.

sector

The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle.

arc

Part of a circle connecting two points on the circle; could be minor (less than 180 degrees - needs 2 letters to name it) or major (greater than or equal to 180 degrees - needs 3 letters to name it)

Central Angle

An angle whose vertex is the center of the circle; equal to the degree measurement of the intercepted arc it creates

Pythagorean Theorem

a²+b²=c² (c=hypotenuse, only true for right triangles)

Slope formula [between 2 points (x₁,y₁) and (x₂, y₂)]

y2 - y1/ x2-x1

Logarithm Formula: log base x = y

b to the y power equals x

sin

opposite/hypotenuse

cos

adjacent/hypotenuse

tan

opposite/adjacent

Mean

Add all numbers, divide by number of numbers.

Median

The middle number from small to large, choose median.

FOIL

First, Outer, Inner, Last

Number of degrees in n-sided polygon

(n-2)(180)

Percent Formula

is/of = %/100

acute angle

an angle less than 90 degrees but more than 0 degrees

adjacent angles

are a pair of angles with a common vertex and a common side, but no common interior points

area

the number of square units needed to cover a flat surface

circle

a closed plane figure made up of all the points the same distance from the center; radius from center to point on circle (1/2 d); diameter from point to point through center (2r)

circumference

the distance around a circle; C = 2*pi*r; C = pi * d

complementary angles

two angles whose sum is a right angle (90 degrees)

cube

a three-dimensional shape with six square or rectangular sides

diameter

the length of a straight line passing through the center of a circle and connecting two points on the circumference

equilateral triangle

A triangle with all sides congruent.

obtuse angle

an angle between 90 and 180 degrees

parallel lines

coplanar lines that do not intersect

parallelogram

a quadrilateral whose opposite sides are both parallel and equal in length

perpendicular lines

Two lines that intersect to form right angles

polygon

a closed plane figure bounded by straight sides

quadrilateral

a four-sided polygon

right triangle

a triangle with one right angle

supplementary angles

Two angles whose sum is 180 degrees

volume

the amount of 3-dimensional space occupied by an object

Area of a square

A=side²

Area of a circle

A=πr²

Volume of a cube

V=s³

Fahrenheit to Celsius

C = 5/9 (F-32)

Celsius to Fahrenheit

F= 9/5 (C+32)

a³-b³

(a-b)(a²+ab+b²)

a³+b³

(a+b)(a²-ab+b²)

undefined slope

vertical line

zero slope

horizontal line

Infinitely many solutions

when the system of equations has the same slope and y-intercept

One solution

when the system of equations has different slopes and possibly the same y-intercept

No solution

when the system of equations has the same slope but different y-intercepts

Elimination method

solving systems by adding or subtracting equations to eliminate a variable (sometimes requires that one or both equations be multiplied by a LCM)

Solution of the system of linear equations

Any ordered pair in a system that makes all the equations true

Graphing method

Graphing the system of equations and finding the point at which they intersect

Substitution method

Replacing one variable with an equivalent expression containing the other variable

Absolute value equation

A V-shaped graph that points upward or downward

translation

transformation of a graph that is a shift of a graph horizontally, vertically, or both, which results in a graph of the same shape and size, but in a different position.

Area of a sector

x°/360 times (πr²), where x is the degrees in the angle

Radius (Radii)

A segment connecting the center of a circle to any point on the circle

Sn=n(a1+an)/2

arithmetic sum; n is the # of terms

An=a1+(n-1)d

arithmetic series; d is the common difference

An=a₁r^(n-1)

geometric series; r is the common ratio (what one term is multiplied by in order to get the next term)

Sn=a1(1-r^n)/(1-r)

geometric sum; r is the common ratio and cannot equal zero

S=a1/(1-r)

geometric infinite sum

reflection

transformation resulting in a mirror image over a particular line (usually the x-axis or y-axis)

dilation

transformation that changes the size of a graph (stretch or compress)

rotation

transformation that "spins" a graph around a particular point

parabola

the graph of a quadratic equation

cosecant

csc = 1/sin

secant

sec = 1/cos

cotangent

cot = 1/tan

y=asin(b*theta)

a: amplitude; b: # of cycles from 0 to 2pi; period: 2pi/b; starts at zero

y=acos(b*theta)

a: amplitude; b: # of cycles from 0 to 2pi; period: 2pi/b; starts at max

Law of Cosines

c^2 = a^2 + b^2 - 2abcosC; b^2 = a^2 + c^2 - 2accosB; a^2 = b^2 + c^2 - 2bccosA; used with SAS or SSS

Law of Sines

sinA / a = sinB / b = sinC / c; used with AAS or ASA or SSA

45-45-90 triangle

hypotenuse = √2 * leg

30-60-90 triangle

hypotenuse = 2 * shorter leg; longer leg = √3 * shorter leg

to convert from radians to degrees

multiply by 180/pi

to convert from degrees to radians

multiply by pi/180

an angle with a vertex on a circle

creates an intercepted arc that is twice the degree of that angle