Double Inequalities
Sometimes, two inequalities are combined into one. However, you need to be
careful:
If x>3 and x<6, then you can write 3<x<6. But, if x<3 or x>6, you can not write
3>x>6 because that would imply that 3>6 and that would be a false state
Factoring
Works well when the quadratic can easily be factored.
The idea behind factoring is to place the equation into standard form, and then factor the left
hand side into two factors (x-a) and (x-b). The solutions to the equation are then x=a and x=b.
Intermediate Value Theorem
Polynomials are continuous functions which mean that you can't pick up your pencil while
graphing them.
Question:
If at some point, you're below the x-axis, and at another point you're above the x-axis,
and you didn't pick up yo
Polynomial Inequalities
Polynomials are continuous. That means that you can draw them without picking up your pencil
(there's a more rigorous definition in calculus, but that definition will work for us, now). If you're
going to change from being less tha
Polynomials are continuous and smooth everywhere.
A continuous function means that it can be drawn without picking up
your pencil. There are no jumps or holes in the graph of a polynomial
function.
A smooth curve means that there are no sharp turns (lik
The natural base e
As x increases without bound, the quantity (1+1/x)^x will approach the transcendental number e.
The limit notation shown is from calculus. The limit notation is a way of asking what happens to
the expression as x approaches the value sh
Change of Base Formula
One dilemma is that your calculator only has logarithms for two bases on it. Base 10 (log) and
base e (ln). What is to happen if you want to know the logarithm for some other base? Are you
out of luck?
No. There is a change of base
Earthquakes
R = log I
The Richter scale is used to measure the intensity of an earthquake. The actual model is a little
more complex, but it simplifies to the equation shown. R is the magnitude on the Richter scale of
the earthquake. I is the intensity of
Linear Factorization Theorem
A polynomial in one variable of degree n>0 can be factored as
f(x)=an (x-c1) (x-c2) (x-c3) . (x-cn)
Where an is the leading coefficient and each c1 . cn is a real or complex root of the function.
Notice that each factor is a l
Equations involving Radicals
1. Isolate the radical term on one side.
2. Square both sides of the equation. Warning! Squaring is not a one-to-one function and
you may introduce extraneous solutions. Also, don't forget that there is a middle term
when squa
Exponential Decay (decreasing form)
Function
y = C e-kt, k > 0
Features
Asymptotic to y = 0 to right
Passes through (0,C)
C is the initial value
Decreasing, but bounded below by y=0
Notes
Exponential decay and be used to model radioactive decay and de
Fundamental Theorem of Algebra
Every polynomial in one variable of degree n, n > 0, has at least one real or complex zero.
Corollary to the Fundamental Theorem of Algebra
Every polynomial in one variable of degree n, n > 0, has exactly n real or complex z
The old standard form for a parabola was written like any other polynomial, f(x) = ax2 + bx + c, a
0.
We're going to complete the square and place it into a form where the translations are easily
interpreted. This time, instead of dividing through by a,