ACT Math

Mutually exclusive events

A∩B = ∅; For example, rolling an even and odd number at the same time cannot happen

Complement events

Outcomes not in an event (keyword: NOT)

Compound Events (A AND B)

P(A∩B) = P(A) * P(B)

Compound Events (A OR B)

P(A∪B) = P(A) + P(B) - P(A and B)

Conditional Probability

Likelihood of an event occurring given that another event has already happened

Independent events: P(A | B) = P(A)

Dependent events: P(A | B) ≠ P(A)

Arithmetic sequence equation

Geometric sequence equation

Sum of arithmetic sequence

Sum of geometric sequence

Simple interest

Compound interest formula

Continuous compound interest formula

Equation of circle

Equation of ellipse

2a = width, 2b = height; major axis is the longer one

Log properties

Box and whisker plot

Law of sines

Law of cosines

When to use: SSS, SAS

Permutation

Order matters

Combination

Order doesn’t matter

Fundamental counting principle

if one event can occur in m ways and a second event can occur in n ways, then the two events can occur in m × n ways; this rule applies to tasks in a sequence, too

Midpoint formula

average of two points

Slope formula

Distance formula

pythag

Similar vs Congruent

Similar shapes have the same angles and form but may vary in size

Congruent shapes are identical in size and shape

Interior angles of a polygon

Unit circle

Multiplying matrices

# of columns in first matrix has to equal # of rows in second matrix

Function transformations

Trig function transformations

Finding the vertex of a parabola

Quadratic formula + discriminant rules

By 0 real solutions, they mean 2 complex roots

Vertex form

Point slope form

Odds in favor

Number of successes : Number of failures

Special right triangles

Finding triangle similarity

AA: two pairs of corresponding angles are equal (third angle must also be too)

SAS: two pairs of corresponding sides are proportional and their included angles are equal (the angle is formed by those two sides)

SSS: if two triangles have proportional sides

Volume of cylinder

Volume of sphere formula

Volume of cone formula

Trapezoid area formula

average of two base lengths

Expected value

weighted average

Bayes’ theorem

Bayes’ theorem is almost like an inverse of conditional probability:

Conditional probability tells you "How likely is this evidence if my hypothesis is true?"

Bayes’ theorem tells you "Now that I've observed this evidence, how likely is my hypothesis?" It provides a way to update the probability of a hypothesis based on new evidence

You usually have to solve for the denominator

Empirical Rule

68-95-99.7 (for normal/bell distribution)

68% of all values fall within 1 standard deviation of the mean

95% of all values fall within 2 standard deviations of the mean

99.7% of all values fall within 3 standard deviations of the mean

Rational function: holes

Happens when numerator and denominator have a common factor.

Example: both have (x - 4) → x = 4 is a hole

Rational functions: horizontal asymptotes

Compare highest powers

Denom degree > Num degree → y = 0

Denom degree = Num degree → y = ratio of leading coefficients
Num degree > denom degree → no horizontal asymptotes

Cylinder surface area

Tangent line of circles

Slope is perpendicular to the radius that touches the point

Rational number properties

Rational numbers (+, -, *, /) another rational number = rational number

Irrational number properties

Cannot be written as a fraction of two integers