Basic Formulas/Identities Flashcards
Algebra, Geometry and Pre-Calc
Parallel Lines
Lines that have the same slope
Example:
y = 2x + 3
y = 2x - 4
Perpendicular Lines
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
Product Rule (Exponents - Same Base)
am × an = am+n
Quotient Rule (Exponents - Same Base)
am / an = am-n
Negative Exponent Rule
a-m = 1/am
Power of a Power Rule
Exponent Rule
(am)n = amn
Power of a Product Rule (Exponents)
(ab)m = ambm
Power of a Quotient Rule
(a/b)m = am/bm
Fractional Exponent Rule
am/n = n√am
Radians (around a circle)
Full Circle: 2π radians
Half Circle: π radians
Quarter Circle: 𝜋/2 radians
What is a coefficient?
A coefficient is a number that is multiplied by a variable in an algebraic expression
Pythagorean Theorem
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
What is the Sum Property of Limits?
This property works similarly for subtraction (the Difference Property of Limits)
What is the Product Property of Limits?
What is the Quotient Property of Limits?
What is the limit of a function multiplied by a constant?
What is the limit of a composite function?
The outer function must be continuous at the limit of the inner function
What is an irrational number?
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
What is an integer?
A natural number or the negation of a natural number
Think of natural numbers as “counting” numbers.
Zero is also considered an integer
What is a logarithm?
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.
What is a natural log?
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:
lne1 = 0
The constant “e” must be raised to the power of 0 in order to get the value 1
What is the “Difference of Squares?”
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.
Pythagorean Trig Identity (Sine & Cosine)
Double Angle Identities (Cosine)
What does it mean for a function to be continuous?
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.
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.
In mathematics, what is the difference between a combination and a permutation?
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.”
How do you calculate the factorial of a number?
0! is equal to 1
What is the formula for a combination?
Also commonly called “n Choose k” formula.
“n Choose k” formula is used to find the number of ways selecting k things out of n things.
Suppose you want to change the base of a logarithm. How could this be done?
What is an inverse function?
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.
To find the inverse of f(x) = y, simply swap y and x and then solve for y again.
