Princeton Math SAT Prep

Area of a Circle

A=πr²

Area of a Triangle

A=1/2bh

Circle equation in standard form

(x-h)^2+(y-k)^2=r^2

(h,k) is the center

r is the radius

Circumference of a circle

C=2πr or C=πd

Discriminant

b^2-4ac, used to determine the number and type of solutions to a quadratic equation

D>0 means 2 real solutions

D=0 means 1 real solution

D<0 means no real solution

Growth and Decay

when a value increases or decreases by a rate or a multiplier

final amount = original amount(1+rate)^# of changes

final amount = original amount(multiplier)^# of changes

linear graphs: slope-intercept form

y=mx+b

linear graphs: standard form

Ax+By=C

Parabolas: factored form

y=a(x-m)(x-n) where m and n are solutions to the equation

Parabolas: standard form

y = ax^2 + bx + c

Parabolas: vertex form

y=a(x-h)^2+k

(h,k) is the vertex

if a positive, opens up, if negative, opens down

Pythagorean Theorem

a²+b²=c²

quadratic formula

-b±[√b²-4ac]/2a

quadratic: products of the solution

quadratic equation in the form ax^2+bx+c=0

c/a

quadratic: sum of the solutions

quadratic equation in the form ax^2+bx+c=0

-b/a

quadratic: vertex in standard form

standard form: (y=ax^2+bx+c)

-b/2a

radians to degrees

180/π radians

pi radians = 180 degrees

slope of a line in slope-intercept form

m in y=mx+b

slope of a line in standard form

-A/B in Ax+By=C

Slope of a line with 2 given points

rise/run (y2-y1)/(x2-x1)

T=AN

mean/average: Total=AveragexNumber of Things

Arc

Part of the circumference of a circle

At least

greater than or equal to

Box plot

A graph that displays the highest and lowest quarters of data as whiskers, the middle two quarters of the data as a box, and the median

cannot find mean or mode

Distinct

Not the same value

Dot plot

a graphical display of data using dots. Each point represents one value. can determine mean, median, mode, range, and standard deviation

Fractional exponents

indicates a power as well as a root. power over root

Inscribed

one figure inside another figure.

touches edges but does not go outside

integer

All whole numbers (both positive and negative) and zero. not a decimal or fraction

irrational number

a number that can NOT be expressed as a ratio of two integers or as a repeating or terminating decimal - Pi or any square root of an imperfect square are considered irrational. a number that cannot be expressed as a fraction

Line of best fit

a line drawn in a scatter plot to fit most of the dots and shows the relationship between the two sets of data

Linear equation

an equation with at least one variable but no exponents.

MADSPM

to Multiply exponents, Add them; to Divide exponents, Subtract them; when there are Parentheses, Multiply exponents

Negative exponents

take the reciprocal of the result of the positive exponent. 1/x^b=x^(-b)

No more than

less than or equal to

Nonlinear equation

an equation with at least one variable that is raised to the second power or higher.

Number of solutions to a linear equation

0 when the variable term is the same but the constants are different

1 when the variable terms are different

infinitely many when the variable terms and the constants are the same

Order of Operations

PEMDAS (Parentheses / Exponents / Multiply / Divide / Add / Subtract)

origin

the center point in the coordinate plane where the axes meet (0,0)

Prime

a number that only has itself and 1 as factors

probability

number of outcomes that gives you what you want/ total number of possible outcomes

proportion

pairs of values with the same part-to-who;e relationship

radius

straight line from the center of a circle to its edge

half the length of the diameter

rate

relationship between two values

often speed over time or work over time

ratio

a part-to-part relationship between numbers

rational number

a number that can be expressed as a fraction

reciprocal

a fraction flipped upside down

Roots

another term for the solutions to a quadratic equation

scatterplot

a type of graph in which each ppint represents one value

sector

a "pie slice" in a circle

SOHCAHTOA

SIN (Opposite/Hypotenuse) COS (Adjacent/Hypotenuse) TAN (Opposite/Adjacent)

Standard Deviation

a measure of how spread out numbers are around the center

a narrower curve means a smaller standard deviation

a wider curve means a larger standard deviation

stem-and-leaf plot

A way to present data with the units digits in the right-hand column (the leaves) and the other digits in the left-hand column (stem).

System

two or more equations or inequalities, often graphed together

two-way table

a way of showing data with the items sorted two different ways

x-intercept

where a graph intersects the x-axis

y=0

y-intercept

where a graph intersects the y-axis

x=0

two triangles that have equal angles and equal sides

proofs:

SSS

ASA

SAS

AAS

double root

when a graph touches the x-axis and then curves away

seen in a fourth-degree polynomial

equivalent expressions

expressions that have the same value but aren't identical

Line intersecting parallel lines

creates big and small angles which add up to 180 degrees

# of solutions to a system of linear equations

0 when lines are parallel (same slope)

infinitely many when the lines are the same

1 when there are two lines the are not the same or parallel

# of degrees in geometric figures

straight line= 180

triangle= 180

four-sided figure= 360

Pythagorean triples

3,4,5

5,12,13

8,15,17

7,24,25

similar triangles

two triangles that have equal angles and proportional sides

proofs: AAA

SAS

SSS

slopes of parallel lines

equal to each other

slopes of perpendicular lines

negative reciprocals

ballparking

use estimating and rounding to eliminate answers that are too big or too small

Plugging In The Answers (PITA)

1. rewrite the answer choices and label them

2. start with a middle number and work the steps of the question

3. look for something in the question that tells you what must happen

4. Stop when an answer choice works

RTFQ

Read The Final Question

Read the end of the question to see what it's asking for

Write down key words on scratch paper

Solving for an expression

find a way to get the expression without needing to isolate a variable

when to use PITA

Question asks for a specific amount

Answer choices are numbers in order

Tempted to write an algebraic equation

When to Plug In

Questions asks about a relationship between variables or numbers

question includes "equivalent expression" or "in terms of"

answer choices have variables

Word Problem Basic Approach

Read the final question (RTFQ)

Let the answers point the way

work in bite-sized pieces

use POE

Working with overlapping shapes

find a line or angle that the two shaped have in common

Allegiance

A commitment or loyalty to a person, group, or cause, often used in the context of loyalty to a nation.

foster

to promote the growth or development of something, such as an idea, relationship, or skill.

fortify

to strengthen or reinforce something, often used in the context of physical structures or mental resilience.

deference

respectful submission or yielding to the judgment, opinion, or will of another.

dissension

disagreement or conflict, often arising from differing opinions or interests.

auspicious

conducive to success; favorable or promising.

futile

incapable of producing any useful result; hopeless or ineffective.

avant-garde

new and experimental ideas or methods, particularly in the arts and culture.

belies

to contradict or show something to be false; to mask or deceive.

chronicles

a factual written account of important or historical events in the order of their occurrence.

inviolate

free from violation or injury; intact.

consecrated

declared sacred or holy, often through a religious ceremony.