End Behavior (review) and Odd/Even Degree of Polynomial Func

 $x\rightarrow-\infty;\ f\left(x\right)\rightarrow?$  

 $x\rightarrow\infty;\ f\left(x\right)\rightarrow?$  

 $f\left(x\right)=3x+1=3x^1+1$  

 $f\left(x\right)=2x^3+3x^2+2x-1$  

 $f\left(x\right)=7x^5-4x^3+2x-8$  

 $f\left(x\right)=x^9$  

All of the degrees of these functions, namely 1, 3, 5, 9, etc. are ODD

The graphs of these functions will have end behavior "at odd" with each other, or pointing in different directions.

 $f\left(x\right)=3x^2+1$  

 $f\left(x\right)=2x^4+3x^3+2x-1$  

 $f\left(x\right)=7x^6-4x^3+2x-8$  

 $f\left(x\right)=x^8$  

All of the degrees of these functions, namely 2, 4, 6, 8, etc. are EVEN

The graphs of these functions will have end behavior pointing in the same direction.

 $f\left(x\right)=3x+1$  

 $f\left(x\right)=2x^3+3x^2+2x-1$  

 $f\left(x\right)=7x^5-4x^3+2x-8$  

 $f\left(x\right)=x^9$  

Notice that the degree is still odd!!!!!  All of the signs of the lead coefficients, namely 3, 2, 7, and 1 are POSITIVE. 

The graphs of these functions will still have end behavior "at odd" with each other, or pointing in different directions.

BUT, they will be INCREASING (so down left, up right).

 $f\left(x\right)=-3x+1$  

 $f\left(x\right)=-2x^3+3x^2+2x-1$  

 $f\left(x\right)=-7x^5-4x^3+2x-8$  

 $f\left(x\right)=-x^9$  

Notice that the degree is still odd!!!!!  All of the signs of the lead coefficients, namely -3, -2, -7, and -1 are NEGATIVE. 

The graphs of these functions will still have end behavior "at odds" with each other, or pointing in different directions.

BUT, they will be DECREASING (so up left, down right).

 $f\left(x\right)=3x^2+1$  

 $f\left(x\right)=2x^4+3x^3+2x-1$  

 $f\left(x\right)=7x^6-4x^3+2x-8$  

 $f\left(x\right)=x^8$  

Notice that the degree is still even!!!  All of the signs of the lead coefficients, namely 3, 2, 7, and 1 are POSITIVE.

The graphs of these functions will have end behavior pointing in the same direction.  But, they will both point up!

 $f\left(x\right)=-3x^2+1$  

 $f\left(x\right)=-2x^4+3x^3+2x-1$  

 $f\left(x\right)=-7x^6-4x^3+2x-8$  

 $f\left(x\right)=-x^8$  

Notice that the degree is still even!!!  All of the signs of the lead coefficients, namely -3, -2, -7, and -1 are NEGATIVE.

The graphs of these functions will have end behavior pointing in the same direction.  But, they will both point DOWN!