ACT FORMULA SHEET MATH

NUMBER AND QUANTITY FUNCTIONS

Sequences Function Notation

Sequence with a constant of c: fog(x) = f(g(x))
arithmetic sequence: an = a1 + c(n – 1) note: this is different from fg(x), which is f(x) × g(x)
geometric sequence: an = a1(c)n – 1
Trigonometry on the Coordinate Plane
Logarithms π radians = 180°
logab = c — ac = b

Rates
distance = speed × time STATISTICS AND PROBABILITY
variation: quantity = rate × time

Percents
percent = part / whole × 100%
ALGEBRA % symbol = divide by 100
a% of b = a/100 × b = a × b/100
percent change = change / original
Linear Graphs
Statistics
y average (mean) = sum of terms / number of terms
median = middle number (or average of 2 middle numbers)
mode = most common number

(x y range = biggest – smallest

Probability and Counting Techniques

probability = number of desired terms / total number of terms
x
O
P(event happens) + P(event doesn’t happen) = 1

(x y P(A and B) = P(A) × P(B)

GEOMETRY

slope-intercept formula: y = mx +b
m = slope = (y1 – y2) / (x1 – x2) Lines and Angles

b = y-intercept = (0,b) supplementary angles add to 180°

distance formula: √[(y1 – y2)² + (x1 – x2)²] complementary angles add to 90°
vertical angles are congruent
midpoint formula: midpoint = [(x1 + x2)/2, (y1 + y2)/2]
Triangles
Quadratics
angles of a triangle add to 180°
FOIL (First, Outer, Inner, Last): (a + b)(c + d) = ac + ad + bc + bd
area of a triangle = ½ × base × height
quadratic formula: X= -b +√b2 - 4ac
2a Pythagorean theorem: a² + b² = c²
discriminant: b² – 4ac
If . . . 45 : 45 : 90 triangle ratio: x : x : x√2
• discriminant > 0 — 2 real solutions 30 : 60 : 90 triangle ratio: x : x√3 : 2x
• discriminant = 0 — 1 real solution
• discriminant < 0 — no real solutions
 y

45°
x=h

x √2 y = a(x−h)2 + k
x
x
O

45°
x
(h, k)
Polygons
trapezoid area = average of bases × height = (b1 + b2)/2 × h
perimeter = sum of sides
sum of angles in n-sided figure = (n – 2) × 180° 3D Figures
area of a rectangle = length × width surface area of rectangular prism: 2(length × width + length
area of a parallelogram = base × height × height + width × height)

Circles and Parabolas volume of rectangular prism: V = lwh

circle arc length = central angle / 360° × circumference volume of right cylinder: V = πr²h

circle sector area = central angle / 360° × area

Triangles and Trigonometry
area of circle = πr²
SOH-CAH-TOA:
circumference of circle = 2πr
diameter of circle = 2r
B

y 3
5

C A
(x-h)2 +(y-k)2 =r2 4

Sine (sin) Cosine (cos) Tangent (tan)

opposite adjacent opposite
hypotenuse hypotenuse adjacent

3 4 3
x 5 5 4
0 (h,k)
sin(x) = opposite / hypotenuse
r cos(x) = adjacent / hypotenuse
tan(x) = opposite / adjacent
sin²(x) + cos²(x) = 1
sin(x) = cos(90 – x)
cos(x) = sin(90 – x)

radius = r
(x – h)² + (y – k)² = r²
center of circle = (h, k)
Vertex form for a parabola:
y = a(x – h)² + k
vertex = (h, k)
axis of symmetry: x = h