Families of Functions with Excel : y = 1/x
Presented by: Charmaine Cureton
CTSE 5040/6040, Write Up #2
October 26, 2007
Problem: We will be looking at the basic graph of y = 1/x. From this we will then see how changes to the basic equation can effect the graph. We will then relate how this ties into transformations of graphs. We will then explore how the same effects will transform the equation y = sin(x).
Getting Started: Lets first examine the behavior of the graph y = 1/x. To do this in Microsoft Excel you could use the chart below to create this graph.
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| What do you notice about the graph? Take a look at the graph when x=0. What happens? Take a look at what happens when the graph gets close to y=0. Does y ever equal zero? Why or why not? | When looking at the graph and the table, one should notice that as the graph gets closer to y=0, it never actually touches y=0. This is because y cannot equal zero. There is not a value for x that will yield zero as a solution for y. When looking at x=0, one should notice that there is not a value for y in the table nor on the graph. This is because when you place x=0 into the equation y=1/x you will yield y=1/0 and you cannot divide by zero. This creates a vertical asymptote at x=0. |
Now lets take a look at how changing the elements of the equation will change the graph.
| Transformations (shifting right and left) | Transformations (compression and stretch) | The Sine Function and transformations |
DISCUSSION:
In my investigation, I explored various constraints could affect the behavior
and graph of the function y = 1/x. My constraints were
• Adding and subtracting a constant k to the variable x. (y = 1/(x+k))
• Adding and subtracting a constant k to the equation. (y = (1/x) +k)
• Multiplying a constant k to the equation. (y = k * (1/x))
Students should be able to recognize that these constraints relate back to transformations
of functions. If you know your properties of transformations then one should
be able to predict the graph of the function. The goal for this lesson is to
help students understand how simple changes to a function can alter its appearance.
Adding and subtracting a constant k, to the variable x will shift the entire
graph to the left and the right. If k is positive, then the graph will shift
k units to the left. If k is negative, then the graph will shift k units to
the left. They should also not the changes of the asymptotes to the graph.
Adding and subtracting a constant k to the equation will shift the entire graph
k units upward or downward. If k is positive then the basic graph will shift
k units upward. If k is negative, the graph will shift k units downward. They
should notice that one asymptote changes but not both.
Multiplying a constant k to the equation will make a compression or a stretch
to the graph. It can also flip the graph across the x axis. If k is negative
then the graph will flip across the x axis. If k>1 then a vertical stretch
will occur. If k is 0 < k < 1 then a vertical compression will occur.
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