The UltimateFormula Sheet for ACT Math

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The Ultimate Formula Sheet for ACT Math

The ACT does not provide any formulas. Be prepared by making sure to have these ones memorized.

Fractions, Decimals, & Percentages: (for this section, r is the percent in decimal form)

part

Fraction

whole

=

;

100

part

Percent =

Percent Increase or Decrease:

100%

old new

old

−

×

Increase by a percent: multiply by

(1 )r+

Decrease by a percent: multiply by

(1 )r−

Simple Interest:

(1 )A P rt= +

Interest Compounded Annually:

(1 )

t

AP r= +

Interest Compounded n times per year:

1

nt

r

AP

n



= +





Rates, Ratios, & Proportions:

General form of a conversion factor:

_

ending units

starting units







Example:

12

10 120

1

inches

feet inches

foot



=





(Concentration of A x Volume of A)

+ (Concentration of B x Volume of B)

= Final concentration (Vol. of A + Vol. of B)

Distance = Rate x Time

Exponents, Roots, & Polynomials:

Multiplication Rule for Exponents:

b c bc

aa a

+

⋅=

Division Rule for Exponents:

b

bc

c

a

−

=

Power Rule for Exponents:

( )

c

b bc

aa=

Negative Exponents:

1

b

a

−

=

Fractional Exponents:

( )

b

c

b

c

a a or a=

1i = −

;

2

1i = −

;

3

ii= −

;

4

1i =

4

1

n

i =

;

41n

ii

+

=

;

42

1

n

i

+

= −

;

43n

ii

+

= −

Complex Conjugates:

( )( )

a bi a bi+−

Parabolas:

Standard Form:

2

()f x ax bx c= ++

;

vertex=

,

22

bb

f

aa



−−





;

y-intercept = c;

x-intercepts =

2

4

2

b b ac

a

−± −

Sum of solutions =

b

a

−

Discriminant =

2

4b ac−

; Pos=2 real roots

Zero= 1 real root; Neg=2 imaginary roots

Factored Form:

( ) ( )( )fx axmxn=−−

;

x-intercepts are m and n;

x-coordinate of vertex =

2

mn+

Vertex Form:

2

() ( )fx ax h k

=−+

;

vertex =

(,)hk

Difference of Squares:

22

( )( )a b a ba b−=+ −

Sum of Cubes:

33 2 2

( )( )a b a b a ab b+=+ −+

Difference of Cubes:

33 2 2

( )( )a b a b a ab b

−=− ++

Perfect Square Trinomial:

( )

2

22

2a ab b a b+ +=+

and

(

)

2

22

2a ab b a b− +=−

Completing the Square:

22

2

22

bb

x bx x

  

++ =+

  

  

Graphing Lines:

Slope Formula:

21

yy

m

xx

−

=

−

Slope of horizontal line = 0

Slope of vertical line = undefined

Standard Form:

Ax By C+=

Slope-Intercept Form:

y mx b= +

Point-Slope Form:

11

()y y mx x−= −

Distance Formula:

22

21 21

( )( )d xx yy

= − +−

Midpoint Formula:

1 21 2

,

22

x xy y

M

++



=





Parallel lines: equal slopes

⊥ Lines: slopes are opposite reciprocals

Parent Graphs & Transformations:

yx=

2

yx=

3

yx

=

x

ya=

yx=

Transformation

Visual effect

()fx k+

Shift up by k units

()fx k−

Shift down by k units

()fx h+

Shift left by h units

()fx h−

Shift right by h units

()fx−

Reflect over the x axis (flip upside down)

()cf x

Stretch vertically by a factor of c (becomes skinnier)

1

()fx

c

Shrink vertically by a factor of c (becomes fatter)

Data & Probability:

__

sum of items

Average

number of items

=

Median = center data point

Mode = most frequent data point

Range = maximum – minimum

_

desired outcomes

Probability

possible outcomes

=

Probability that independent events A and B will

both happen:

( ) ( ) ()PA B PA PB= ×

Probability that either A or B will happen:

()()()()PA B PA PB PA B=+−

Expected Value:

1

() ( )

n

ii

i

Ex x Px

=

∑



Angles:

Vertical ∠’ s are ≅

∠’s that form a linear pair are supplementary (add

up to 180°)

∠’ s that form a circle add up to 360°

When ∥ lines are cut by a transversal, all acute ∠’ s

are ≅ and all obtuse ∠’ s are ≅

Triangles:

Area of a Triangle:

1

2

A bh=

The three ∠’ s of a ∆ add up to 180°

An exterior ∠ is equal to the sum of the two

remote interior ∠’ s

Pythagorean Theorem:

222

abc+=

Pythagorean Triples: 3-4-5 and 5-12-13

Special Right Triangles:

Circles:

Area of a Circle:

2

Ar

π

=

Circumference of a Circle:

2Cr

π

=

A radius and tangent make a

right ∠

A central ∠ is double the

inscribed ∠

360

x arc

circumference

=

and

sec

360 _ _

x tor

area o f circle

=

where x = central angle

Polygons: (for this section, n is the number of sides)

Area of a Rectangle:

A lw=

Area of a trapezoid:

12

1

()

2

b bh

+

Sum of the exterior angles: 360°

Sum of the interior angles:

180( 2)n −

One int. ∠ of a regular polygon:

180( 2)n

n

−

# of diagonals:

( 3)

2

nn−

(convex only)

Properties of Parallelograms:

1. Opp sides are ∥ and ≅

2. Opp ∠’ s are ≅

3. Consec ∠’s are supplementary

4. Each diagonal forms a pair of ≅∆’s

5. Diagonals bisect each other

If they are ≅ it is a rectangle

If they are ⊥ it is a rhombus

6.

Area base height= ×

Solids:

Volume of a Rectangular Prism (Box):

V lwh=

Surface Area of a Box:

2( )SA lw lh wh= ++

Volume of a Cylinder:

2

V rh

π

=

Surface Area of a Cylinder:

2

22

SA r rh

ππ

= +

Volume of a Sphere:

3

4

3

Vr

π

=

Volume of a Cone:

2

1

3

V rh

π

=

Volume of a Pyramid:

1

3

V lwh=

Trigonometry:

sin

opp

hyp

=

cos

adj

hyp

=

tan

opp

adj

=

1

csc( )

sin( )

x

=

1

sec( )

cos( )

x

=

1

cot( )

tan( )

x

=

360°=2π radians

sin

tan

cos

x

=

22

sin cos 1xx+=

sin( ) cos(90 )xx= −

Law of Sines:

sin sin sin

abc

ABC

= =

Law of Cosines:

2 22

2 cos( )

a b c bc A=+− 

sin( )yx=

cos( )yx=

tan( )yx=

If

sin( )y A Bx C D= −+

(also for cos, csc, and sec)

Amplitude:

A

Period:

2

B

π

Phase Shift:

C

B

Vertical Shift: D

If

tan( )y A Bx C D= −+

(also for cot)

Amplitude: none Period:

B

π

Phase Shift:

C

B

Vertical Shift: D

Sequences and Series: where

1

a

= first term, n = number of terms, d = common difference, r = common ratio

Arithmetic sequence:

1

( 1)

n

aa n d=+−

Sum of an arithmetic series:

( )

1

2

nn

n

S aa= +

Geometric sequence:

1

n

a ar

−

=

Sum of a geometric series:

1

( 1)

1

n

ar

S

r

−

=

−

Logarithms:

If

log

b

ax

=

, then

x

ba=

log

b

a

b

=

Vector Addition:

22

2 cos

a b a b ab

θ

+= + +



Matrix Multiplication: Only possible when columns of first = rows of second

A B E F AE BG AF BH

C D G H CE DG CF DH

++

    

×=

    

++

    

Determinant of

AB

AD BC

CD



= −





Conic Sections:

Circle:

2 22

( )( )xh yk r

− +− =

, where (h,k) is the center and r is the radius

Ellipse:

22

( )( )

1

xh yk

ab

−−

+=

where (h,k) is the center, 2a is the horizontal axis, and 2b is the vertical axis

Horizontal Ellipse:

2 22

abc= +

Vertical Ellipse:

2 22

bac= +

where c is the distance from center to focus

Horizontal Hyperbola:

22

( )( )

1

xh yk

ab

−−

−=

Vertical Hyperbola:

22

( )( )

1

yk xh

ab

−−

−=