Private Prep: "Need to Know" Math Facts
Important Number Facts
Exponent Rules
1
x · x = xa+b
x−n = n
x
xa
√
a
a−b
=
x
b
x = b xa
xb
32 = 9
(xa )b = xab 22 = 4
a
b
Integer: "whole" numbers — positive,
negative, or zero.
Zero: neither positive nor negative.
Zero is even.
x1 = x
23 = 8
33 = 27
x0 = 1
24 = 16
34 = 81
Translation
3—4—5
6—8—10
5—12—13
8—15—17
7—24—25
Primes: 1 isn't prime. 2 is the only
even prime.
Mean, Median, Mode
is means equals
of means multiply
per means divide
cent means 100
difference means
subtract
what or a
number means x
Pythagorean
Theorem
a 2 + b2 = c 2
Triples
Special Right Triangles
30°—60°—90°
sum of terms
# of terms
note: # of terms · mean = sum of terms
Mean (average):
Median: middle number in ordered
list (or mean of center terms)
45◦
s
30◦
2x
45°—45°—90°
√
s 2
√
x 3
45◦
60◦
Mode: most common number
x
s
Shapes and Formulas
Circle
Rectangle
Triangle
Parallelogram
l
r
tangent
h
w
h
h
d
b
A = πr2
C = 2πr = πd
Equation of a Line
slope-intercept:
slope:
-intercept:
y
x-intercept:
slope:
y = mx + b
m
b
−b/m
∆y
y2 − y 1
rise
=
=
run
∆x
x2 − x1
point-slope: (y − y1 ) = m(x − x1 )
A = lw
P = 2l + 2w
Transformations
y
y
y
y
= f (x) + b : up b
= f (x) − b : down b
= f (x + b): left b
= f (x − b): right b
Parabolas
y = a(x − h)2 + k
vertex: (h, k)
axis: x = h
Factoring Patterns
Difference of Two Squares (D.O.T.S.):
(x − y)2 = x2 − 2xy + y 2
x2 − y 2 = (x + y)(x − y) (x + y)2 = x2 + 2xy + y 2
b
A=
b
bh
2
A = bh
Triangle Inequality
The length of any side in a triangle is
between the absolute value of the
difference and the sum of the other two
sides, i.e.,
|a − b| < c < a + b
for sides a, b, and c.
Midpoint
Distance
The x (or y) value
of the midpoint is
the average of the
x (or y) value of two
points.
To find the distance
between two points,
draw a right triangle
and use the Pythagorean theorem.
Private Prep: ACT Math Facts
Trigonometry Basics
hy
opposite
po
ten
Identities
sin2 x + cos2 x = 1
us
A
C
Trigonometric Laws
a
SOH-CAH-TOA
B
b
adj
cos A =
hyp
opp
tan A =
adj
c
Quadrant II
180 degrees = π radians
Students
only sin > 0
π
radians
180
180
radians = r ·
degrees
π
Law of sines:
Law of Cosines: c2 = a2 + b2 − 2ab cos C
π
2
θ
1
Calculus
only cos > 0
Quadrant III
Quadrant IV
3π
2
Complex Numbers
Logarithms
a
b =x
i0 = 1
2π
−1
π
2
1
π
i =i
3π
2
−1
−
1
cos x
i=
√
π
2
−1
i2 = −1
i3 = −i
4
i =1
logb xy = logb x + logb y
x
logb = logb x − logb y
y
logb xy = y logb x
logb x = a
log x = log10 x
logb b = 1
logb 1 = 0
logb bn = n
Important Definitions
tan x
π
2
All
sin > 0
cos > 0
tan > 0
Take
only tan > 0
sin x
3π
2
0/2π
π
2π
π
Quadrant I
(cos θ, sin θ)
d degrees = d ·
π
2
a
b
c
=
=
sin A
sin B
sin C
A
Unit Circle
r
sin x
cos x
e
adjacent
opp
sin A =
hyp
tan x =
amplitude: half vertical distance
between "crest" and "trough."
even function: f (−x) = f (x)
( sin x is even.)
period: length over which a function odd function: f (−x) = −f (x)
( cos x is odd.)
repeats. (E.g. sin x has period 2π .)
Equation of a Circle
(x − h)2 + (y − k)2 = r2
center: (h, k)
radius: r
Quadratic Formula
ax2 + bx + c = 0
√
−b ± b2 − 4ac
x=
2a
Polynomials
Sequences and Series
If (x − a) is a
factor of the
polynomial p(x) ,
then p(a) = 0 .
arithmetic sequence: a, a + d, a + 2d, . . . (common difference)
nth term: a + (n − 1)d
1
sum: a + (a + d) + · · · + (a + (n − 1)d) = (a + an )n
2
If p(a) = 0 , then
(x − a) is a factor
of p(x) .
geometric sequence: a, ar, ar2 , ar3 , . . . (common ratio)
nth term: arn−1