Math 118
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2
Function
-A & B are 2 nonempty sets. -A function from A to B is a rule of correspondence that assigns to each element in set A to exactly one element in B.
Properties of Logs
-LOGbP + LOGbQ = LOGbPQ -LOGbP - LOGbQ =LOGb P/Q -LOGbP^r = rLOGbP -b^(LOGbP) = P
Polynomial Function
-n is a nonnegative integer -ai's are constants -if an does not equal 0 the degree of the polynomial function is n, the largest exponent on the input variable
Compound Interest Formula
A + P(1+r/n)^t p= principle investment r= rate of interest t=time (yrs) n= how many times interest compounded /yr A= new amount
interest compounded continuously
A = Pe^rt
complex numbers
All numbers, a combination of a real and an imaginary numbers.
Change of Base
LOGaX = (LOGbX)/(LOGbA)
Doubling Time Formula
T2 = ln2/K k= growth constant
Natural Numbers
The basic counting numbers
Real Numbers
any number that can be expressed in decimal form
Rational Numbers
can be written as a fraction of integers (0 cant be a dominator)
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Linear Function
f(x) + Ax+B
Quadratic Function
f(x) = ax^2 + bx + c
y=√x
graph goes from origin, right. curving up and out straight
y=x^2
graph goes up on both sides like a U
y = |x|
graph goes up on both sides like a V
y=x^3
graph goes up on right side like a U then down on left side, so its a curving line
y=1/x
graph is 2 asymptotes, reflecting about the origin
y-√[a^2 - x^2]
graph is a semi circle
e
irrational number approx. 2.7 as a base for exponential functions is most useful
Root of equation
is a number when substituted for x leads to a true statement. f(r)=0 so we also refer to the number as a zero of the function
Exponential Function with Base B
let b denote an arbitrary positive constant other than 1. (b>0, b cannot equal 1) y=b^x
Fundamental Theorem of Algebra
let f(x) be a polynomial of degree 1 or greater. The fundamental theorem of algebra asserts that the equation f(x)=0 has at least one root among complex numbers
The Factor Theorem
let f(x) be a polynomial. If f(r)=0 then x-r is a factor of f(x). Also, if x-r is a factor of f(x) then f(r)=0
ln
ln means LOGeX
Integers
natural numbers, zero and negatives
Irrational Numbers
numbers that when written in decimal form, the decimal has no pattern & does not terminate ( ex: pie)
The Division Algorithm
p(x) = d(x) * q(x) + R(x) -px and dx are polynomials -dx is not the zero polynomial -qx and rx are unique polynomials -rx is either the zero polynomial or the degree of rx is less than the degree of dx
Exponential Decay
refers specifically to decrease or decay governed by functions of the form y=ae^bx (where a is positive and b is negative)
Exponential Growth
refers specifically to growth governed by functions of the form y=ae^bx (a&b both positive constants)
Absolute Value Function
the absolute value of a real number x, denoted by |x|, is the distance from x to the origin
LOGbX
the exponent to when b must be raised to yield x ex: LOG2-8 =3 -> 2^3=8
Half-life
the half life of a radioactive substance is the time required for half of a given sample to disintegrate. The half-life is an intrinsic property of the substance; it does not depend on the given sample size.
Asymptote
the line continually approaches 0 but never touches the axis
Remainder Theorem
when a polynomial f(x) is divided by x-r the remainder is f(r)
Vertex Formula
x = -b/2a
Quadratic Equation
x= (-b +/- √[b^2 -4ac])/2a
Rational Function
y = f(x) / g(x)
Vertex in Completing the Square
y=(x-1)^2 +2 vertex: (1,2)
