End Behavior of Polynomials, Exponentials, and Logarithms

Presentation

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Mathematics

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11th Grade - University

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Practice Problem

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Hard

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CCSS

HSF-IF.C.7E

Standards-aligned

Bo Gilbert

Used 8+ times

FREE Resource

13 Slides • 6 Questions

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

x\ \rightarrow\ \infty

f\left(x\right)\ \rightarrow\ \infty

Odd Degree, Negative Leading Coefficient

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

x\ \rightarrow\ \infty

f\left(x\right)\ \rightarrow\ -\infty

Even Degree, Positive Leading Coefficient

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

x\ \rightarrow\ \infty

f\left(x\right)\ \rightarrow\ \infty

Even Degree, Negaative Leading Coefficient

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

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.

End Behavior of Polynomials, Exponentials, and Logarithms

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