Basic Formulas/Identities

Question 1

Parallel Lines

Answer 1

Lines that have the same slope

Example:
y = 2x + 3
y = 2x - 4

Question 2

Perpendicular Lines

Answer 2

Lines where the slopes are negative reciprocals of each other. Perpendicular lines form a 90 degree angle when they intersect.

Example:
y = 2x + 3
y = -1/2 + 2

Question 3

Product Rule (Exponents - Same Base)

Answer 3

a^m × aⁿ = a^m+n

Question 4

Quotient Rule (Exponents - Same Base)

Answer 4

a^m / aⁿ = a^m-n

Question 5

Negative Exponent Rule

Answer 5

a^-m = 1/a^m

Question 6

Power of a Power Rule

Exponent Rule

Answer 6

(a^m)ⁿ = a^mn

Question 7

Power of a Product Rule (Exponents)

Answer 7

(ab)^m = a^mb^m

Question 8

Power of a Quotient Rule

Answer 8

(a/b)^m = a^m/b^m

Question 9

Fractional Exponent Rule

Answer 9

a^m/n = ⁿ√a^m

Question 10

Radians (around a circle)

Answer 10

Full Circle: 2π radians
Half Circle: π radians
Quarter Circle: 𝜋/2 radians

Question 11

What is a coefficient?

Answer 11

A coefficient is a number that is multiplied by a variable in an algebraic expression

Question 12

Pythagorean Theorem

Answer 12

The Pythagorean Theorem can be used to find the unknown side of a right-angled triangle when two other sides are known.

a² + b² = c²

Question 13

What is the Sum Property of Limits?

Answer 13

This property works similarly for subtraction (the Difference Property of Limits)

Question 14

What is the Product Property of Limits?

Answer 14

Question 15

What is the Quotient Property of Limits?

Answer 15

Question 16

What is the limit of a function multiplied by a constant?

Answer 16

Question 17

What is the limit of a composite function?

Answer 17

The outer function must be continuous at the limit of the inner function

Question 18

What is an irrational number?

Answer 18

A real number that cannot be expressed as a fraction or ratio of two integers

Essentially, it’s a number with a decimal representation that goes on forever

Question 19

What is an integer?

Answer 19

A natural number or the negation of a natural number

Think of natural numbers as “counting” numbers.

Zero is also considered an integer

Question 20

What is a logarithm?

Answer 20

A logarithm is a mathematical function that tells you the exponent to which a base number must be raised to get a specific result.

For example:
log₁₀100 = 2

10 (the base) must be raised to the power of 2 to get the value 100.

Question 21

What is a natural log?

Answer 21

A natural logarithm, denoted as “ln,” is a logarithm where the base is the mathematical constant “e” (approximately 2.718), meaning it tells you what power you need to raise “e” to in order to get a specific number.

Example:
ln_e1 = 0

The constant “e” must be raised to the power of 0 in order to get the value 1

Question 22

What is the “Difference of Squares?”

Answer 22

The difference of two squares is a squared number subtracted from another squared number. Every difference of squares may be factored according to the identity in elementary algebra.

This is often helpful when rationalizing fractions.

Question 23

Pythagorean Trig Identity (Sine & Cosine)

Answer 23

Question 24

Double Angle Identities (Cosine)

Answer 24

Question 25

What does it mean for a function to be continuous?

Answer 25

A function is continuous if, over a given interval, the limit of the function as x approaches every x-value in the domain (except for left/right boundaries) is equal to the value of the function at that x-value. ## Footnote The left/right boundaries can also be included as well, but for the left side of the interval the function must equal the limit as x approaches from values greater than x, whereas for the right side of the interval the function must equal the limit as x approaches from values less than x.

Question 26

In mathematics, what is the difference between a combination and a permutation?

Answer 26

A "*combination*" refers to selecting a group of items where the order of selection doesn't matter, while a "*permutation*" refers to selecting a group of items where the order of selection does matter; essentially, a "*permutation*" is an ordered "*combination*."

Question 27

How do you calculate the factorial of a number?

Answer 27

## Footnote 0! is equal to 1

Question 28

What is the formula for a combination? ## Footnote Also commonly called "n Choose k" formula.

Answer 28

## Footnote "n Choose k" formula is used to find the number of ways selecting k things out of n things.

Question 29

Suppose you want to change the base of a logarithm. How could this be done?

Answer 29

Question 30

What is an inverse function?

Answer 30

An inverse function in math is a function that "undoes" the operations of another function, meaning if a function *f(x)* takes input x and produces output y, then the inverse function of *f* would take input y and produce the original input x back again; essentially reversing the process of the original function. ## Footnote To find the inverse of *f(x) = y*, simply swap y and x and then solve for y again.