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)
Fraction =
Simple Interest: =
A P(1 + rt )
part
part
; Percent =
whole
100
Percent Increase or Decrease:
old − new
old
Interest Compounded Annually: =
A P(1 + r )t
×100%
Increase by a percent: multiply by (1 + r )
Interest Compounded n times per year:
r

A P 1 + 
=
n


nt
Decrease by a percent: multiply by (1 − r )
Rates, Ratios, & Proportions:
General form of a conversion factor:
 ending _ units 


 starting _ units 
 12inches 
Example: 10feet 
 = 120inches
 1foot 
(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: ab ⋅ ac =
a b +c
Division Rule for Exponents:
ab
= a b −c
ac
( )
Power Rule for Exponents: ab
c
= abc
1
Negative Exponents: a = b
a
b
Fractional Exponents: a c = c ab or
i=
( a)
c
b
−1 ; i 2 = −1 ; i 3 = −i ; i 4 = 1
i 4 n = 1 ; i 4 n+1 = i ; i 4 n+2 = −1 ; i 4 n+3 = −i
Complex Conjugates: (a + bi )(a − bi )
−b
Parabolas:
Standard Form: f ( x ) = ax 2 + bx + c ;
Discriminant = b2 − 4ac ; Pos=2 real roots
 b  b 
vertex=  − , f  −   ;
 2a  2a  
Zero= 1 real root; Neg=2 imaginary roots
y-intercept = c;
x-intercepts are m and n;
x-intercepts =
−b ± b2 − 4ac
2a
Sum of solutions =
−b
a
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Factored Form: f ( x ) =a( x − m)( x − n) ;
x-coordinate of vertex =
m+n
2
Vertex Form: f ( x ) = a( x − h)2 + k ;
vertex = (h, k )
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Difference of Squares: a2 − b2 =(a + b)(a − b)
Sum of Cubes: a3 + b3 =(a + b)(a2 − ab + b2 )
Difference of Cubes: a3 − b3 =(a − b)(a2 + ab + b2 )
Perfect Square Trinomial: a2 + 2ab + b2 =( a + b ) and a2 − 2ab + b2 =( a − b )
2
2
b
b 
Completing the Square: x 2 + bx +   =  x + 
2
2 
2
2
Graphing Lines:
Slope Formula: m =
y 2 − y1
x2 − x1
Standard Form: Ax + By =
C
Slope-Intercept Form:=
y mx + b
Slope of horizontal line = 0
Point-Slope Form: y − y1 = m( x − x1 )
Slope of vertical line = undefined
Distance Formula: d=
( x2 − x1 )2 + (y 2 − y1 )2
 x + x y +y 
Midpoint Formula: M =  1 2 , 1 2 
2 
 2
Parallel lines: equal slopes
⊥ Lines: slopes are opposite reciprocals
Parent Graphs & Transformations:
y=x
y= x
y = x2
The Ultimate Formula Sheet for ACT Math
y = x3
y = ax
y= x
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Transformation
f (x) + k
f (x) − k
f ( x + h)
f ( x − h)
−f ( x )
cf ( x )
Visual effect
Shift up by k units
Shift down by k units
Shift left by h units
Shift right by h units
Reflect over the x axis (flip upside down)
Stretch vertically by a factor of c (becomes skinnier)
1
f (x)
c
Data & Probability:
Average =
Shrink vertically by a factor of c (becomes fatter)
sum _ of _ items
number _ of _ items
Median = center data point
Mode = most frequent data point
Probability that independent events A and B will
both happen: P( A =
B) P( A)× P(B)
Probability that either A or B will happen:
P( A  B) = P( A) + P(B) − P( A  B)
n
Range = maximum – minimum
Probability =
Expected Value: E( x ) = ∑ xi P( xi )
i =1
desired _ outcomes
possible _ outcomes
Angles:
Vertical ∠’ s are ≅
∠’s that form a linear pair are supplementary (add
up to 180°)
Triangles:
1
Area of a Triangle: A = bh
2
∠’ 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 ≅
Special Right Triangles:
The three ∠’ s of a ∆ add up to 180°
An exterior ∠ is equal to the sum of the two
remote interior ∠’ s
Pythagorean Theorem: a2 + b2 =
c2
Pythagorean Triples: 3-4-5 and 5-12-13
Circles:
Area of a Circle: A = π r 2
A radius and tangent make a
right ∠
The Ultimate Formula Sheet for ACT Math
Circumference of a Circle: C = 2π r
A central ∠ is double the
inscribed ∠
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x
arc
=
360 circumference
x
sec tor
=
360 area _ of _ circle
and
where x = central angle
Polygons: (for this section, n is the number of sides)
Area of a Rectangle: A = lw
Area of a trapezoid:
One int. ∠ of a regular polygon:
1
(b1 + b2 )h
2
# of diagonals:
Sum of the exterior angles: 360°
180(n − 2)
n
n(n − 3)
(convex only)
2
Sum of the interior angles: 180(n − 2)
Properties of Parallelograms:
1. Opp sides are ∥ and ≅
5. Diagonals bisect each other
3. Consec ∠’s are supplementary
If they are ⊥ it is a rhombus
If they are ≅ it is a rectangle
2. Opp ∠’ s are ≅
6. Area
= base × height
4. Each diagonal forms a pair of ≅∆’s
Solids:
Volume of a Rectangular Prism (Box): V = lwh
4
Volume of a Sphere: V = π r 3
3
Surface Area of a Box: SA= 2(lw + lh + wh)
1
Volume of a Cone: V = π r 2 h
3
Volume of a Cylinder: V = π r 2 h
Surface Area of a Cylinder:=
SA 2π r 2 + 2π rh
1
Volume of a Pyramid: V = lwh
3
Trigonometry:
sin =
opp
hyp
cos =
360°=2π radians
adj
hyp
tan x =
tan =
opp
adj
sin x
cos x
csc( x ) =
The Ultimate Formula Sheet for ACT Math
sec( x ) =
sin2 x + cos2 x =
1
a
b
c
Law of Sines: = =
sin A sin B sin C
y = sin( x )
1
sin( x )
1
cos( x )
cot( x ) =
1
tan( x )
=
sin( x ) cos(90 − x )
Law of Cosines: a2 = b2 + c 2 − 2bc cos( A)
y = cos( x )
y = tan( x )
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If y A sin(Bx − C ) + D (also for cos, csc, and sec)
=
Amplitude: A
Period:
2π
B
Phase Shift:
C
B
Vertical Shift: D
Phase Shift:
C
B
Vertical Shift: D
If y A tan(Bx − C ) + D (also for cot)
=
Amplitude: none
Period:
π
B
Sequences and Series: where a1 = first term, n = number of terms, d = common difference, r = common ratio
Arithmetic sequence: an = a1 + (n − 1)d
Sum of an arithmetic series:=
Sn
n
( a1 + an )
2
Geometric sequence: an = a1 r n−1
Sum of a geometric series: Sn =
a1 (r n − 1)
r −1
Logarithms:
If log b a = x , then b x = a
 
Vector Addition: a + b=
log b a =
log a
log b
a2 + b2 + 2ab cosθ
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 
A B
Determinant of 
=
 AD − BC
C D
Conic Sections:
Circle: ( x − h)2 + (y − k )2 =
r 2 , where (h,k) is the center and r is the radius
Ellipse:
( x − h)2 (y − k )2
+
=
1 where (h,k) is the center, 2a is the horizontal axis, and 2b is the vertical axis
a2
b2
2
2
Horizontal Ellipse: a=
b2 + c 2 Vertical Ellipse: b=
a2 + c 2 where c is the distance from center to focus
Horizontal Hyperbola:
( x − h)2 (y − k )2
(y − k )2 ( x − h)2
Vertical
Hyperbola:
−
=
1
−
=
1
a2
b2
a2
b2
The Ultimate Formula Sheet for ACT Math
©2020, Test Prep for Success LLC