ACT Math

Challenge areas for ACT math including equations, definitions, etc.

logbase(n)=x (logarithmic)

base^x=n (exponential)

y=A sin (B(x-h)) + k AND y= A cos(B(x-h)) +k

• A is the amplitude (when A < 0, the graph will be reflected vertically over the midline

• B is the frequency in the 2pi interval, and the period can be determined by -2pi/B (when B<0, the graph will be reflected horizontally)

• h<0 shifts the function left, h>0 shifts the function right (phase/horizontal shift)

• k<0 shifts the function (and thus the mean/midline) down, k>0 shifts the function (and thus the mean/midline) up

Double angle identities

• sin(2A)=2sinAcosA

• cos(2A)=cos^2A-sin^2A

Sum of angles of a polygon

(n - 2) * 180degrees

Distance formula (finds distance between two points on a graph)

d=square root of ((x1-x2)² + (y1-y2)²)

Midpoint formula (finds midpoint between two points on a graph)

(Xm, Ym) = ((x1+x2)/2, (y1+y2)/2)

Parallelogram

a flat, four-sided geometric shape where both pairs of opposite sides are parallel

Area of a trapezoid

A=(b1+b2)/2(height)

Area of a rhombus

A=(d1)(d2)/2 d=diagonals

Circumference of a circle

Pi(diameter) AND =2pi(radius)

Area of a circle

Pi(radius)^2

Equation of a circle (center = (h,k) and radius = r)

(x-h)² +(y-k)² = r²

Volume of a cylinder

pi(radius)^2(height)

Volume of a rectangular prism

lwh

Volume of a cube

e^3 (e=edge length)

Surface area of a cube

6e^2 (e=edge length)

Volume of a sphere

(4/3)pi(radius)^3

Law of sines

sinA/a = sinB/b + sinC/c ABC=angles, abc = sides

Law of cosines

A²=B²+C²-2BCcos(a) (Can also be B² or C²)

Formula for combinations (Use when you select a smaller group of items from a larger set and the order does not matter, EX: Team or Committee Selection: Picking 4 students out of 20 to join a club.)

n! / (r!(n-r)!) n=set of items/groups r = some items/groups

Negative Exponent

Make the original number into a fraction with 1/n^x (n=original number) and (x=exponent)

Fractional exponent

The denominator tells you the root to take, and the numerator tells you the power to raise the base to. ex: a^m/n = m square root of (a²)

Square roots

You can multiply and divide square roots but cannot add or subtract them unless base numbers are the same.

Inverses

Find by switching the independent and dependent variables (aka x and y) and then solving for the dependent variable again.

Pythagorean Identities

sin^2 + cos^2 =1, 1+tan^2=sec^2, 1+cot^2=csc^2

Unit circle

2pi (1,0), pi/6 ((square root of 3/2), ½), pi/4 ((square root of 2/2)/(square root of 2/2)), and pi/3 (same as pi/6 but x and y are switched)

30-60-90 triangle

short leg of x, a hypotenuse of 2x, and a long leg of x(square root of 3)

45-45-90 triangle

legs of x and a hypotenuse of x(square root of 2)

Cosecant

csc =1/sin (hyp/opp) reciprocal of sin

Secant

sec=1/cos ((hyp/adj) reciprocal of cosine

Cotangent

cot=1/tan (adj/opp) reciprocal of tangent

For independent events

P(A|B)=P(A), P(B|A)=P(B), and P(A and B)=P(A) * P(B)

Conditional Probability rule

P(A|B)=P(A and B)/P(B)

Use this when you want to find the probability of an event happening given that another event has already occurred. It narrows down your total possibilities.

ex: In a class, 30% of students play soccer, and 15% play both soccer and basketball. If you randomly pick a student who you already know plays soccer, what is the probability they also play basketball? P(soccer = 0.30) P(Soccer and basketball)=0.15… 0.15/0.30 = 50%

Addition Rule (Non-Mutually exclusive)

P(A or B)=P(A)+P(B)-P(A and B)

Multiplication Rule (independent events)

P(A and B) = P(A) * P(B)

Use this to find the probability of Event A AND Event B happening together, when the first event does not change the odds of the second event.

Complement rule

P(not A) = 1-P(A)

Use this when it is easier to calculate the probability of an event not happening than the event itself

General Multiplication Rule (Dependent Events)

P(A and B) + P(A)*P(B|A)

Use when you want two events to happen in a row, but the first action changes the total pool for the second action.

ex: Picking marbles without replacement: A bag has 3 red and 2 blue marbles. You pick two without putting the first back. What is the probability of picking Red AND then another Red?

Addition Rule (Mutually Exclusive)

P(A or B) = P(A) + P(B)

how many ___ are possible

option 1 * option 2 *option 3 ex: 5 people, 5 seat choices: 5 times 4 times 3 times 2 times 1 = 12 possible seating arrangements

P(A and B) vs P(A or B)

multiplication vs addition

SD

measure of how spread data is (data close or far from mean)

Volume of a prism

area of base times height (area of base = area of a circle)

area of a circle

(pi)r²

circumf. of a circle

2pi(r)