ACT Math Formulas

a°

1

a¹

a

a^-m

1/(a^m)

a^1/n

ⁿ|a

a^m/n

ⁿ|a^m

a^m * aⁿ

a^m+n

a^m/aⁿ

a^m-n

(a^m)ⁿ

a^mn

aⁿbⁿ

(ab)ⁿ

aⁿ/bⁿ

(a/b)ⁿ

Definition of a Logarithm

If log_ab=c, then a^c=b.

Change of Base Rule

log_ab = (log b)/(log a)

log_m(ab)

log_ma+log_mb

log_m(a/b)

log_ma - log_mb

log_ma^b

b log_m a

Common Difference

d=a_n+1-a_n

nth term

a_n=a₁+(n-1)d

Sum of the first nth terms

S_n=n/2*(a₁+a_n)

Common Ratio

r=(a_n+1)/a_n

nth term

a_n=a₁r^n-1

sum of the first \(n\) terms of a finite geometric series

S_n=a₁*(1-rⁿ)/(1-r)

Difference of Two Squares

a²-b²=(a-b)(a+b)

Perfect Square Trinomial

(a+/-b)²=a²+/-2ab+b²

Sum & Difference of Cubes

a³+/-b³=(a+/-b)(a²+/-ab+b²)

Complex Number

a+bi

Value of i

i=|-1 i² =-1

√-n

i|n

(a+bi)(a-bi)

a²+b²

Sum of n Consecutive Integers

n/2(a₁+a_n)

Sum of n Consecutive Odd Integers

n/2(a+l)

Sum of Natural Numbers (Sequence begins from 1)

n(n+1)/2

x intercept

y=0, crosses x axis

y intercept

x=0, crosses y axis

Distance

d=|(x₂-x₁)²+(y₂-y₁)²

Midpoint

p = (x₁+x₂)/2, (y₁+y₂)/2

Linear Equation

y=ax+b

Quadratic Equation

y=ax²+bx+c

Cubic Equation

y=ax³+bx²+cx+d

Absolute Value Equation

y=|x-h|+k

Finding the Slope

m=(y₂-y₁)/(x₂-x₁)

Slope Intercept From

y=mx+b

Point-Slope Formula

(y-y₁) =m(x-x₁)

Quadratic Function Equations

y=ax²+bx+c

y=a(x-h)²+k

Quadratics

a<0
a>0

Opens Downward

Opens Upward

Quadratic Formula for x

x=(-b±|b²-4ac)/2a

Vertex Formula (h,k)

-(2a/b), (4ac- b²)/4a OR -(b/2a., f(h))

Discriminate

b² - 4ac

D>0

2 Real Solutions

D=0

1 Real Solution

D<0

2 Imaginary Solutions

Part Percentage

Part = (percent/100)*whole

Whole Percentage

Whole = part/(part/100)

Percent

(part/whole)*100%

Permutation

n!/(n-r)!

Combination

n!/r(n-r)!

Mean

sum of terms/number of terms

Range

max value - min value

Mode

Most frequent number to appear

Median

Central number when list is arranged from least to greatest

Probability

desired outcomes/total number of outcomes

Complementary Angles

<a+<b=90

Supplementary Angles

<a+<b=180

Parallel Lines

a=d=e=h

b=c=f=g

c+e=180,d+f=180

Intersecting Lines

a=b,c=d

a+c=180

b+d=180

Sum of Angles in a Triangle

<a+<b+<c=180

Area of Triangle

1/2b*h

Exterior Angle Theorem

<a+<b=<c

Similar Triangle Theorem

a/d = b/c = c/f

cos x

sin(90-x)

sin x

cos(90-x)

csc x

sec(90-x)

sec x

csc(90-x)

cot x

tan(90-x)

tan x

cot(90-x)

Trig Quadrant Signs

All Students Take Calculus

Area of a Trapezoid

A=(a+b)/2*h

Area of Parallelogram

A=bh

Interior Angles

(n-2)(180)

Exterior Angle

=360

Circumference of a Circle

2pieR/pie*D

Area of a Circle

pie * r²

Equation of Circle

(x-h)²+(y-k)²=2

Arc Length

n/360 * 2pier

Sector Area

n/360 * pier²

Volume of Cube

V=s³

Cube Surface Area

6s³

Volume of Sphere

V=3/4 pie*r³

Surface Area of Sphere

4pie*r²

Volume of Cone

V=1/3 pie*r²h

Surface Area of Cone

pie*r²+pie*r*l

Volume of Cylinder

V = pier²h

Surface Area of Cylinder

2pie*r²+ 2*pie*r*h

Rectangular Prism Area

V=lwh

Rectangular Prism Area

2(lw+wh+lh)