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

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

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"End behavior" describe the behavior of f(x) (the y-values) as x-values get infinitely smaller (more negative) or infinitely larger (more positive)

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

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

"End behavior" can be described as...

Up/down right and up/down left OR

right up/down and left up/down

To describe end behavior, focus on the arrows on the left side of the y-axis and the right side of the y-axis

The arrows appear in blue on this graph.

Only the arrows; nothing else matters...

Like that. There's an arrow on the left, or when  $x\rightarrow-\infty$  and an arrow on the right, or when  $x\rightarrow\infty$  

End behavior...

The arrow on the left points down, or  $f\left(x\right)\rightarrow-\infty$  and the arrow on the right points up, or  $f\left(x\right)\rightarrow\infty$  .

Sometimes, the arrow goes down on the left and up on the right

That's called increasing...

Sometimes, the arrows go up on the left and down on the right...

That's called decreasing...

Sometimes both arrows go up

But nothing in between matters...

Sometimes both arrows go down

But nothing in between matters.

The question we have to ask ourselves is...

What kind of clues can we gather from the function to indicate the end behavior?

The answer is two-fold:

The degree of the function (the highest exponent of its variable) and the sign of the lead coefficient (the number that multiplies the variable with the largest exponent)

Let's examine the degree first

•  $f\left(x\right)=3x+1=3x^1+1$  

•  $f\left(x\right)=2x^3+3x^2+2x-1$  

•  $f\left(x\right)=7x^5-4x^3+2x-8$  

•  $f\left(x\right)=x^9$  

• 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.

Multiple Choice

The degree of the following function is... (careful)

 $f\left(x\right)=-4x^4+2x^3-3+3x^5$  

• odd

• even

This is what all of the graphs of these functions, with ODD degrees, look like...

The left and right arrows point in different directions. NOTHING in between matters.

Multiple Choice

The ends of the following function will point...

 $f\left(x\right)=-4x^4+2x^3-3x+3x^5$  

• in the same direction

• in opposite directions

Let's continue to examine the degree

•  $f\left(x\right)=3x^2+1$  

•  $f\left(x\right)=2x^4+3x^3+2x-1$  

•  $f\left(x\right)=7x^6-4x^3+2x-8$  

•  $f\left(x\right)=x^8$  

• 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.

Multiple Choice

The degree of the following function is... (careful)

 $f\left(x\right)=-4x+3x^2+2$  

• even

• odd

This is what all of the graphs of these functions, with EVEN degrees, look like...

The left and right arrows point in the same direction. NOTHING in between matters.

Multiple Choice

The end behavior of the functions will point...  (careful)

 $f\left(x\right)=-4x+3x^2+2$  

• in different directions

• in the same direction

Let's examine the effect of the sign of the lead coefficient

•  $f\left(x\right)=3x+1$  

•  $f\left(x\right)=2x^3+3x^2+2x-1$  

•  $f\left(x\right)=7x^5-4x^3+2x-8$  

•  $f\left(x\right)=x^9$  

• 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).

Multiple Choice

The lead coefficient of this function is... (careful)

 $f\left(x\right)=-3x^2+3x^4-8$  

• positive

• negative

This is what all of the graphs of these functions, with ODD degrees, and POSITIVE lead coefficient look like...

The left and right arrows point in different directions with down arrow on the left and up arrow on the right. NOTHING in between matters.

Multiple Choice

This function over all will be... (careful)

 $f\left(x\right)=-3x^2+3x^4-8$  

• increasing

• decreasing

What if we make the lead coefficient negative? 

•  $f\left(x\right)=-3x+1$  

•  $f\left(x\right)=-2x^3+3x^2+2x-1$  

•  $f\left(x\right)=-7x^5-4x^3+2x-8$  

•  $f\left(x\right)=-x^9$  

• 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).

Multiple Choice

The lead coefficient of this function is... (careful)

 $f\left(x\right)=3x^3-2x^4+3$  

• negative

• positive

This is what all of the graphs of these functions, with ODD degrees, and NEGATIVE lead coefficient look like...

The left and right arrows point in different directions with up arrow on the left and down arrow on the right. NOTHING in between matters.

Multiple Choice

This function overall will be...  (careful)

 $f\left(x\right)=3x^3-2x^4+3$  

• decreasing

• increasing

Let's continue to examine the effect of the sign of the lead coefficient

•  $f\left(x\right)=3x^2+1$  

•  $f\left(x\right)=2x^4+3x^3+2x-1$  

•  $f\left(x\right)=7x^6-4x^3+2x-8$  

•  $f\left(x\right)=x^8$  

• 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!

Multiple Choice

The lead coefficient of this function is... (careful)

 $f\left(x\right)=-3x^3-2x^2+5x^4$  

• negative

• positive

This is what all of the graphs of these functions, with EVEN degrees and POSITIVE lead coefficient, look like...

The left and right arrows point in the same direction, which is up. NOTHING in between matters.

Multiple Choice

Overall this function will have end behavior that...

 $f\left(x\right)=-3x^3-2x^2+5x^4$  

• both point up

• both point down

What if we make the lead coefficient negative? 

•  $f\left(x\right)=-3x^2+1$  

•  $f\left(x\right)=-2x^4+3x^3+2x-1$  

•  $f\left(x\right)=-7x^6-4x^3+2x-8$  

•  $f\left(x\right)=-x^8$  

• 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!

Multiple Choice

The lead coefficient of this function is... (careful)

 $f\left(x\right)=3x^3-2x^2-5x^4$  

• negative

• positive

This is what all of the graphs of these functions, with EVEN degrees and NEGATIVE lead coefficients, look like...

The left and right arrows point in the same direction and will be pointing down. NOTHING in between matters.

Multiple Choice

Overall this function will have end behavior where the ends...

 $f\left(x\right)=3x^3-2x^2-5x^4$  

• both point up

• both point down