End Behavior of Polynomials, Exponentials, and Logarithms

End Behavior of Polynomials, Exponentials, and Logarithms

Review Plus Quiz

End Behavior of Polynomials

• First, check (or determine) the highest degree.

• Second, check (or determine) the sign of the coefficient of the highest degree.

• There are 4 possibilities, which will be explored in the next few slides.

Odd Degree, Positive Leading Coefficient

As  $x\ \rightarrow\ -\infty$   ,  $f\left(x\right)\ \rightarrow\ -\infty$  .

As  $x\ \rightarrow\ \infty$  ,  $f\left(x\right)\ \rightarrow\ \infty$  .

Odd Degree, Negative Leading Coefficient

As  $x\ \rightarrow\ -\infty$   ,  $f\left(x\right)\ \rightarrow\ \infty$  .

As  $x\ \rightarrow\ \infty$  ,  $f\left(x\right)\ \rightarrow\ -\infty$  .

Even Degree, Positive Leading Coefficient

As  $x\ \rightarrow\ -\infty$   ,  $f\left(x\right)\ \rightarrow\ \infty$  .

As  $x\ \rightarrow\ \infty$  ,  $f\left(x\right)\ \rightarrow\ \infty$  .

Even Degree, Negaative Leading Coefficient

As  $x\ \rightarrow\ -\infty$   ,  $f\left(x\right)\ \rightarrow\ -\infty$  .

As  $x\ \rightarrow\ \infty$  ,  $f\left(x\right)\ \rightarrow\ -\infty$  .

End Behavior of Exponential Functions

• First, identify the base b.

• If b > 1, then the following behavior occurs:

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ 0$  .

• As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  .

End Behavior of Exponential Functions

• If  $0\ <\ b\ <\ 1$  , then the following behavior occurs:

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  .

• As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ 0$  .

End Behavior of Logarithmic Functions

• First, identify the base, b.

• If b > 1, the following occurs:

• As $x\ \rightarrow\ 0,\ f\left(x\right)\ \rightarrow\ -\infty$  .

• As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  .

• *Note: Recall that x can only be positive for  $f\left(x\right)\ =\ \log_b\left(x\right)$  .

End Behavior of Logarithmic Functions

• First, identify the base, b.

• If 0 < b < 1, the following occurs:

• As $x\ \rightarrow\ 0,\ f\left(x\right)\ \rightarrow\ \infty$  .

• As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  .

• *Note: Recall that x can only be positive for  $f\left(x\right)\ =\ \log_b\left(x\right)$  .

Now that we have had some review, let's practice some problems.

Multiple Choice

What is the end behavior of the function

 $f\left(x\right)\ =\ 2x\left(x\ -\ 4\right)\left(x\ \ +\ \ 1\right)\$  ?

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

Multiple Choice

What is the end behavior of the function

 $f\left(x\right)\ =\ 3^x$  ?

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ 0$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ 0$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

Multiple Choice

What is the end behavior of the function

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

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

Multiple Choice

What is the end behavior of the function

 $f\left(x\right)\ =13x^6\ -\ 12x^5\ -\ 7x^2\ -\ 18$  ?

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

Multiple Choice

What is the end behavior of the function

 $f\left(x\right)\ =\ \log_{\frac{1}{2}}\left(x\right)$  ?

• As  $x\ \rightarrow\ 0,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ -\infty$  
  As  $x\ \ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

• As  $x\ \rightarrow\ 0,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ -\infty$  

• As  $x\ \rightarrow\ -\infty,\ f\left(x\right)\ \rightarrow\ \infty$  
  As  $x\ \rightarrow\ \infty,\ f\left(x\right)\ \rightarrow\ \infty$  

Poll

How confident do you feel with the end behavior of functions?

• I understand this well.

• I just need a bit more practice.

• This is difficult,, but I understand it a little.

• This is a topic I am struggling with.