We note that x ^ -1 = 1 / (x ^ 1) = 1 / x.

• Thus when x = -2, y = 1 / x = 1 / (-2) = -1/2.
• The y values corresponding to x = -2, -1, 1 and 2 on the table below are obtained in a similar manner.
• For the moment we won't discuss the middle three points.

If we plot the points found so far, we obtain the 'blue' points on the graph marked NO!.

• We see that as the x values increase along the positive x axis, the y values are going to decrease, though they will never reach 0.
• Thus the positive x axis will be a horizontal asymptote for the function.
• In a similar manner we see that the negative y axis is also a horizontal asymptote for the function.

A common error in graphing this function for the first time is to suppose that y = 0 when x = 0.

• In fact, as will see shortly, 1/0 is undefined; and in any case it is certainly not 0.
• However, having made this error is not unreasonable to complete the graph as with the red segment between (-1,-1) and (1,1).

The reason this is an error is not hard to see if we think about what division means.

• For example, 100 / 20 can be visualized by thinking of a line 100 units long, divided into segments 20 units long.
• Clearly there are 5 such segments, so 100 / 20 = 5.
• Alternatively we could count to 100 by 20s.
• We would again quickly see that 100/20 = 5.

If we divide 1 by .1, we see that it takes 10 segments of length .1 to make one segment of length 1, so the quotient is 10.

• Of course we could count to 1 by .1:    .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.
• Again we see that the quotient is 10.

If we divide 1 by smaller number it is clear that are result will be larger, and that since there is no limit to how small number can be there must be no limit to how large or result can be.

Now if we go all the way to 0 with our small divisor, things break down on us.

• The reason is that we can't count to anything by 0's.
• To demonstrate, we could start counting: 0, 0, 0, 0, ... .
• Go ahead and finish. Call me when you are done.

From the above discussion we see that as x gets close to 0, x will go into 1 more and more times.

• So that as we move closer to the y axis (where x = 0) with our graph, our result becomes larger and larger, without limit.
• Thus the graph has vertical asymptotes to the right and to the left of the y axis.
• To the right the vertical asymptote will be along the positive y axis, corresponding to division by positive x values. To the left of the y axis the vertical asymptote will be in the negative direction, corresponding to division by small negative values of x.

A table and graph of y = x ^ -2 are shown in the next figure.

• We note that by squaring the values of x, we obtain all positive values of y.
• We note also that as a result the table and the graph are symmetric about x = 0 (on the graph, about the y axis).

We first introduce the symbol "delta", which is the capital Greek letter which he eventually evolved into capital D.

• Capital 'delta' is shaped like an equal lateral triangle, though it doesn't much look like one in the picture.
• This symbol stands for "change in".
• So when we want to refer, for example, to the change in y, we would put a delta in front of the y, as depicted below.
• Since there is currently no reliable way to put a delta symbol into an Internet document, we will write `d to stand for the delta symbol.

The table of y vs. t will be assumed to represent part of a depth vs. clock time table.

• We have already learned to calculate the average rate which the depth changes between two data points by dividing the change in depth by the change in clock time.
• For example between the first two data points the change in the depth is -20 cm, which takes place during a clock time difference, or time interval, of 10 seconds.
• This gives us an average rate of depth change of -2 cm/second.

Using the delta notation, we can thus say that `dy = - 20 cm (remember to think of `dy and 'delta' y, has written below), and that `dt = 10 sec, so `dy / `dt = - 20 cm / 10 sec = - 2 cm / sec.

We now consider what happens if we graph these points and calculate the slopes from point to point.

• From the first point, at clock time 0 and depth 100 cm, we descend to the second point at clock time 10 seconds and depth 80 cm, moving along the red line from point to point.
• As you know, the slope of any line segment is found by dividing its rise by its run: slope = rise/run.
• The rise from the first point to the second is thus seen to be -20 cm, and the run is 10 seconds.

• These numbers should look familiar from our previous set of calculations.

• The rise represents the change in the y coordinate, or the change in the depth of the fluid.
• The run represents the change in the t coordinate, which is the time interval over which the change in depth takes place.

When we calculate the slope we divide the rise by the run.

• In this case we will be dividing - 20 cm by 10 sec, obtaining the result -20 cm / 10 sec = -2 cm / sec.
• It should be completely clear that we have done the same set of calculations we did before when we calculated the average rate at which the depth changes.