### SEQUENCES 2

#### S. Kastberg

It looks as though there are many correspondences around when we consider

 x 1 2 4 8 y -2 -1 0 1 2 3

In particular it looks like and that .

So now let us consider what happens when How about

Now how about Can you you generalize what happens?

What numbers can m and n be? What numbers can you get when you multiply x terms? Can you multiply any two positive real numbers together to get an x value and find a y value to which it corresponds? Can every positive number be written as 2 to some power?

What other properties does the correspondence between x and y have?

Are these properties also true when values of x are increased by a product of b and values of y are increased by a sum of a, where a and b are positive real numbers? Let's investigate.

 x 1 y -2a -1a 0 1a 2a 3a

Now what is the correspondence between y and x? What properties does it have?

Now lets look at a graph that you are familiar with. Click HERE to investigate the area under the curve using GSP.

Can you develop a table of values like the one above with the heights of the areas as the x values and the total area as the y-value? Are the properties of the correspondence the same for this data as the data that we generated above?

Reflecting on patterns

How do the properties of the correspondence change when b changes?

Does each value of correspond to only one value of a?

What values of b are possible? What values of b are not possible?

What values of a are possible?

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