Graphing polynomial functions

Hi guys its Caylene, for my blog im going to review graphs of polynomial functions.


When we graph polynomials we look at the x-intercepts, y-intercept, the degree, and sign of the leading coefficients.

Example:                        f(x) = 2x³ - 3x² - 11x + 6


first we list our possible zeros:  +/- 1 +/-2 +/-3 +/-6

Next we use remainder theorem:   f(x) = 2(1)³ - 3(1)² - 11(1) + 6
                                                             2 - 3 - 11 + 6 =   -6


x-1 is not a factor because it does not equal to zero

lets try again:                                f(x) = 2(3)³ - 3(3)² - 11(3) + 6
                                                         2(27)  -  3(9) - 33 + 6
                                                          54 - 27 - 33 + 6 = 0


This works so x-3 is a factor because it equals to zero.

Next we go to synthetic division:      



3  2    −36−1196−6−−−−−−−−−−−−−−−−23−20

Now we have :

2x3−3x2−11x+6x−3=2x2 + 3x−2

How did we do this synthetic division? 

Step 1 : Write down the coefficients of the polynomial p(x). Put the zero from x−3=0 ( x=3 ) at the left.

3  2    −3    −11    6    −−−−−−−−−−−−−−−−            

Step 2 : Bring down the leading coefficient to the bottom row.

3  2    −3    −11    6    −−−−−−−−−−−−−−−−2         

Step 3 : Multiply by the number on the left, and carry the result into the next column: 3⋅2=6

3  2    −36−11    6    −−−−−−−−−−−−−−−−2         

Step 4 : Add down the column: −3+6=3

3  2    −36−11    6    −−−−−−−−−−−−−−−−23      

Step 5 : Multiply by the number on the left, and carry the result into the next column: 3⋅3=9

3  2    −36−1196    −−−−−−−−−−−−−−−−23      

Step 6 : Add down the column: −11+9=−2

3  2    −36−1196    −−−−−−−−−−−−−−−−23−2   

Step 7 : Multiply by the number on the left, and carry the result into the next column: 3⋅(−2)=−6

3  2    −36−1196−6−−−−−−−−−−−−−−−−23−2   

Step 8 : Add down the column: 6+(−6)=0

3  2    −36−1196−6−−−−−−−−−−−−−−−−23−20

Bottom line represents the polynomial quotient (2x2+3x−2) with a remainder of 0.

Then it factors even further                        

                                                                    2x² + 3x - 2

 These are our zeros --------------->        (x-3) (x+2) (2x-1)

To find the y intercept we plug 0 into our equation. 

  f(x) = 2x³ - 3x² - 11x + 6
  f(x) = 2(0)³ - 3(0)² - 11(0) + 6
  f(x) = 6

 Y intercept is 6

  

Time to Graph

As we can see our zeros or x intercepts are - 2, 1/2, and 3

Our Y intercept is 6

To find out how far our curve goes we take the 2 ans plug it into our equation
 
 f(x) = 2x³ - 3x² - 11x + 6
  f(x) = 2(2)³ - 3(2)² - 11(2) + 6
  = - 12 

this would be the E2 in the graph

  f(x) = 2x³ - 3x² - 11x + 6
  f(x) = 2(-1)³ - 3(-1)² - 11(-1) + 6


Behavior of the function : Odd function because its cubed
                                       positive coefficient 
                                       down in quadrant 3 and up in quadrant 1

Y int : 6
Zeros : - 2, 1/2, and 3



Reminder: 
 Even dregree.......

If the leading coefficient is positive both arms go up

If the  leading coefficient is negative both arms go down


Odd degree.....

If the leading coefficient is positive left arm will go down, right will go up

If the  leading coefficient is negative right arm will go down and left will go right



Graphing Rules:

1. Each root of the equation that is unique (only one) - the curve crosses at the x-axis
example : (x + 1) (x - 4) (x + 5)

2. If you have two or more of the same root ..... an even number of the same root the graph will bounce at the x axis.
example: (x+1)²

or if its an odd number of the same root the curve will cross and kind of flatten at x axis
example : (x+1)³

3. when you have really high degrees the graph will flatten out more



I hope this helped you with understanding how to graph polynomial functions

Happy spring break to you guys :)