local and global behavior

(Section 22)

We will use the graphing calculator to explore the local and global behavior of a function. By local behavior, we mean what happens to the y-values as we stay very close to a particular x value?

Enter the equation in your calculator and view it in a Zoom 4. Decimal window.
Let's study this function near zero.
First, you know this function has no function value at zero because of the x in the denominator.
Therefore, even though the function appears to have a value at x=0, there is really a hole in the function when the function is near x = 0. So look at the graph with the axes turned off. (2nd Format (zoom)). Notice there is a whole in the function where the y-axis would have crossed.
What value is the function skipping over? There are two ways to study this. First let's look at a set of table values in a small window around x=0. Let the Tablemin= -.4 and .
Notice that all the y-values are staying very close to 1. Therefore we could say that the point the function is skipping over is (0,1). This could also be viewed by tracing on the function when x is very close to zero.
What does this function do when x gets very large positively or negatively.
To do this you might pick windows with higher values like 100<x<125 and -.1<y<.1
From this graph it appears that the y values bounce back and forth along the x-axis. It is very difficult to tell what the values are doing. Try changing the window to 100<x<200 and -.02<y<.02. From this window you get the idea that the y-values are indeed approaching zero as x gets larger and larger.
To confirm this we could build a table and study the actual y-values. Remember that you are only seeing a few values for (x,y), but just try to confirm what you see going on with the graph.
You should also check the negative values to confirm that the same thing is going on: that as x gets very small the y-values are bouncing back and forth around y=0, but they are getting close as x keeps getting smaller.
What have we learned about the function ? First we know that locally there is a hole in the function at (0,1). The function values approach y=0 as x gets very large and very small.
In Calculus this year we will studying other local behavior about this function, such as it's slope at or -.

Check your understanding.

1. Study the function on your calculator. Set up tables and graphs to determine what values the function is approaching as x approaches zero and as x approaches positive and negative infinity or very large numbers and very small numbers.

Updated on 05/29/10 .