ACT Math

Sequences

Sequence with a constant c

arithmetic sequence

geometric sequence

Arithmetic sequence

a_n = a₁ + c(n-1)

for example, in (3, 7, 11, 15, …), the common difference is +4

Geometric sequence

a_n = a₁(c)^n-1

for example, in (2, 6, 18, 54, …), each step multiplies by 3

Logarithms

tells you what power you need to raise a base number to get another number. For example,

log₁₀(100) =2 —- because 10² = 100

log_a b = c

a^c = b

Function notation

fog(x)

f(g(x))

π radians (degree and radian conversion)

180 degrees

Percents

a% of b

a/100 × b = a × b/100

for example,

• Find 20% of 50

• Method 1: 20/100 times 50 = 10

• Method 2: 50 /100 times 20 = 10

• Both ways give the same answer of 10

percent change

change / original

Statistics

mean (average)

sum of terms / number of terms

median

middle number (or average of 2 middle numbers)

mode

most common number

range

biggest – smallest

Probability

number of desired outcomes / total number of outcomes

P(event happens) + P(event doesn’t happen)

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P(A and B)

P(A) × P(B)

Linear graphs

Slope intercept form

y = mx + b

m (slope)

(y₂-y₁) / (x₂-x₁)

b

y - intercept

distance formula

√( (y₂-y₁)² + (x₂-x₁)² )

Midpoint formula

( [x₁ + x₂] / 2 , [y₁ + y₂] / 2)

Quadratics

discriminant

b² - 4ac in quadratic formula

if

• discriminant > 0, 2 solutions

• discriminant = 0, 1 solution

• discriminant < 0, 0 solutions

Lines and angles

Supplementary

angles add up to 180

Complementary

angles add up to 90

Vertical Angles

Triangles

Pythagorean

a² + b² = c²

45 - 45 - 90

30 - 60 - 90

Polygons

Trapezoid Area

height * (base₁ + base₂ ) / 2

Parallelogram Area

base * height

Sum of angles in n-sided figure

(n - 2) * 180 degrees

Circles

Circle Area

πr²

Circumference

2πr

Arc Length

(central angle / 360) * circumference

Sector Area

(central angle / 360) * area

Circle Center

(h,k)

r²

(x-h)² + (y-k)² = r²

any point (x, y) that's exactly distance r from the center (h, k) is on the circle. That's literally the distance formula squared — (x−h)2+(y−k)2(x-h)^2+(y-k)^2 (x−h)2+(y−k)2 is the squared distance from (x,y) to (h,k), and setting it equal to r2r^2 r2 forces every point on the circle to be exactly r away from center.

Parabolas

3D figures

Surface area of rectangular prism

2 (length * width + length * height + width * height)

Volume of a right cylinder

πr²h

Trigonometry

sin²(x) + cos²(x)

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