Rules of Logs

Change of Base Formula
                           
                           
Properties of Logarithms

                          

                          
                          
Examples

                         
                         
Using The Properties

Rational Functions:

A rational function can be written as:

f(x) = N(x)

         D(x)

where N(x) and D(x) are polynomials and D(x) is not a zero polynomial. Every polynomial is a rational function.

To graph a rational function you have to find

- x-intercept

- y-intercept

- vertical asymptote

- horizontal asymptote

- the vertical asymptotes are the zeros of the denominator

- the graph has a horizontal asymptote if the degrees of the polynomials are the same. The horizontal asymptote is the ratio of the coefficients. 

To graph the function 

f(x) =          x                

        - x - 2

   

- the x-intercept is (0,0)

- the y-intercept is (0,0)

- the vertical asymptotes are x = -1 and x = 2 

- no horizontal asymptote

Polynomial Functions of Higher Degree

Polynomial Functions and their end-behavior:

When looking at the graphs of Any polynomial function of nth degree, there are two main things to note.
1) The number of x-intercepts= n
2) The number of Relative Extrema = n-1
Relative Extrema: Relative minimums and maximums

End Behavior:
When predicting the end behavior for a polynomial graph, you must look at the leading term/ highest power of x.

• If nth is even, both arrows will point in the same direction.
• If in is odd, the arrows will face in opposite directions.

Example:

Direction of End Behavior:

Right Hand End behavior: To find the direction of the right hand end behavior, you must look at the sign of the leading term.

Left hand end behavior: To find the direction of the left hand end behavior, you look at the exponent of the leading term to see if its arrow will point in the same direction as the right (even) or the opposite direction (odd). 

Example: Find the end behavior of both left and right sides of the polynomial function

The right hand end behavior: The arrow will point down on the right side because the sign on the leading term is negative (-3).

The left hand end behavior will also be pointing down because x is raised to an even power

Polynomial Functions (Completing the Square)

Polynomial Functions:

Defined as in the form:



Restrictions:
- n must be a non negative integer
- Coefficiants must be real
- Powers of x must be whole numbers
- The degree of the polynomial is the greatest value of n (where )
- the domain of a polynomial is always all real numbers


Example of a Polynomial Function:

Converting from Standard Form to Vertex Form (Completing the Square):

Standard Form: 

Vertex Form:     

Start with the function 

Move aside the 15                                  

The goal is to turn            into a perfect square

To do so, we need to add 16. Normally this would be done by adding 16 to both sides, but because it is being shown as a function , we must add 16 and subtract 16 to the same side (equaling 0).

                                                    
                                                             
                                 

Example 1:


                   
          
                 



Example 2: