5.8 - End Behavior

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

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 

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  $\uparrow\downarrow$  

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

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 $\downarrow\uparrow$  

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

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

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.