End Behavior of a Function (Using Graphs and Tables)
end behavior of a function
Description/Explanation/Highlights
Video Description of End Behavior of a Function
This video explains end behavior of a function using graphs and tables.
Steps and Key Points to Remember When Determining End Behavior of a Function Using Graphs and Tables
To determine end behavior of a function using graphs and tables, follow these steps:
- End behavior of a function refers to observing what the y-values do as the value of x approaches negative as well as positive infinity.
- As a result of this observation, one of three things will happen. First, as x becomes very small or very large, the value of y will approach \(-\infty\). Secondly, it may approach \(\infty\). Finally, it may approach a number.
End Behavior of a Function Where Both Move toward Negative Infinity
- In the graph above, as the value of x gets smaller (to the left), the value of y also get smaller. However, as the value of x gets larger (to the right), the value of y gets smaller in contrast.
- As a result, we write: \(x\rightarrow-\infty \ y\rightarrow -\infty \\ x\rightarrow\infty\ y\rightarrow -\infty\)
End Behavior Where One Moves Toward Negative Infinity, One Toward Positive Infinity
- As x gets smaller (moving toward negative infinity) in the graph above, similarly the value of y also gets smaller. In the same way, as the value of x gets larger (moving toward infinity), likewise, the value of y gets larger.
- Therefore, we write this as: \(x\rightarrow-\infty \ y\rightarrow -\infty \\ x\rightarrow\infty\ y\rightarrow \infty\)
End Behavior of a Function Where One Moves Toward Positive Infinity, One Toward Negative Infinity
- If we reflect the previous graph over the y-axis, as a consequence, the end behavior is reversed. Therefore, as x gets smaller, y now gets larger. and as x gets larger, y, at the same time, gets smaller.
- As a result, we write: \(x\rightarrow-\infty \ y\rightarrow \infty \\ x\rightarrow\infty\ y\rightarrow -\infty\)
End Behavior Where One Moves Toward Negative Infinity, One Toward a Number
- Finally, notice how in the graph above the values of y go toward negative infinity as the values of x move in the same direction. However, this time as x gets larger, the graph begins to flatten out and actually approach the x-axis or \(y=0\).
- We write this as: \(x\rightarrow-\infty \ y\rightarrow -\infty \\ x\rightarrow\infty\ y\rightarrow 0\)
- In the graph above, it is pretty easy to tell that the y-values are approaching 0 as x increases. However, to be sure (or if the number is impossible to tell accurately by looking at the graph), use a table of values to determine the endpoint behavior.
- In the example, \(f(x)=\frac{\textstyle 3x+2}{\textstyle 2x-3}\), it is hard to tell what the graph is approaching when looking at a graph on a calculator. Consequently, we will create a table of large and small values to get a better look.
- First, start with small values of x to see what the graph is doing as x approaches negative infinity. Then do a table of large values to see what y does as x approaches positive infinity.
X
-500
-1000
-1500
-2000
Y
1.4935
1.4968
1.4978
1.4984
X
500
1000
1500
2000
Y
1.5065
1.5033
1.5022
1.5016
- As the values of x become smaller and smaller in the first table, the values of y, as a result, get closer and closer to 1.5 from just below 1.5. Furthermore, as the values of x get larger and larger in the second table, the values of y get closer and closer to 1.5 from just above it.
- As a result, we write this as: \(x\rightarrow-\infty \ y\rightarrow 1.5 \\ x\rightarrow\infty\ y\rightarrow 1.5\)
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Here are some key points to keep in mind when determining endpoint behavior of a function:
- When using a graph look at what the y-values of the graph do as it moves to the far left and far right of the graph.
- The graph will approach \(-\infty\), \(\infty\), or some constant value (number).
- This behavior is determined as x-values increase or decrease to very large or very small values. Likewise, the end behavior is based on what the y-values do.
- When trying to determine end behavior from a table, substitute small and large values for x. Then, look at the results for y.
- The y-values will either continue to increase (approaching \(\infty\)), continue to decrease (approaching \(-\infty\), or they will approach a constant value.
Video Highlights
- 00:00 Introduction
- 00:09 Explanation of what end behavior is and how to determine it
- 01:58 Example of a graph that approaches negative and positive infinity
- 03:02 Another example of a graph approaching both negative and positive infinity
- 03:32 Example of a graph that approaches a constant value (y=0)
- 05:00 \(f(x)=\frac{\textstyle 3x+2}{\textstyle 2x-3}\) example of determining end behavior of a function using a table
- 08:00 Conclusion
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