5.8 - End Behavior

5.8 - End Behavior

You already know that functions have distinctive shapes:

• A linear function graphs as a straight line

• A quadratic graphs as a U-shaped curve that opens up or down

• An absolute value function graphs as a V-shaped curve that opens up or down

End Behavior

• a description of the values of the function as x approaches positive infinity ( $x\rightarrow+\infty$  ) or negative infinity ( $x\rightarrow-\infty$  )

• The degree of the polynomial and its leading coefficient determine the end behavior 

Example: Determine the leading coefficient, degree, and end behavior of the polynomial

 $P\left(x\right)=-4x^3-3x^{2\ }+5x+6$  

• The leading coefficient is -4, which is negative

• The degree is 3, which is odd

• As  $x\rightarrow-\infty,\ P\left(x\right)\ \rightarrow+\infty$  , and as  $x\rightarrow+\infty,\ P\left(x\right)\rightarrow-\infty$  

• Some teachers/videos/books show this as up/down, or  $\uparrow\downarrow$  

How am I ever going to remember all the combinations?

• Think of the easiest cases.

• For a linear function (degree 1, which is odd), if the slope (leading coefficient) is negative the line trends down from left to right. So

   $\uparrow\downarrow$  

• For a linear function (degree 1, which is odd), if the slope (leading coefficient) is positive the line trends up from left to right. So $\downarrow\uparrow$  

How am I ever going to remember all the combinations?

• Think of the easiest cases.

• For a quadratic function (degree 2, which is even), if the a-value (leading coefficient) is negative the graph opens down. So

   $\downarrow\downarrow$  

• For a quadratic function (degree 2, which is even), if the a-value (leading coefficient) is positive the graph opens up. So $\uparrow\uparrow$  

Multiple Choice

Determine the end behavior of the graph of the polynomial

 $x^6-7x^5+x^3-2$  

•  $\uparrow\downarrow$  

•  $\downarrow\uparrow$  

•  $\uparrow\uparrow$  

•  $\downarrow\downarrow$  

Multiple Choice

What do you know about the rule (equation) of the polynomial based on its graph?

• Leading coefficient is positive, degree is even

• Leading coefficient is negative, degree is even

• Leading coefficient is positive, degree is odd

• Leading coefficient is negative, degree is odd

Turning Points

• A turning point is where a graph "turns around" , that is changes from increasing to decreasing or from decreasing to increasing.

• A polynomial function of degree n has at most n-1 turning points, and at most n x-intercepts.

• If the function has n distinct real roots, then it has exactly n-1 turning points and exactly n x-intercepts.

Multiple Choice

How many turning points will a quartic function with four real zeros have?

• 1

• 2

• 3

• 4

Multiple Choice

Which function could describe this graph?

• $f\left(x\right)=-2x^5+x-4$

• $f\left(x\right)=3x^3-9x$

• $f\left(x\right)=-x^3+5x^2+4x+3$

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

Open Ended

What questions do you have for me?