� Assignment One �

**Examine: **

What happens if the 4 is replaced by other numbers (not necessarily integers)? Try 5, 3, 2, 1, 1.1, 0.9, -3.

Let s first examine this equation using values larger than four.

By using values larger than four, we are able to see both curves changing. The curve in Quadrant II extends farther outward (moving up and towards the left). The curve in Quadrant IV extends farther downward (moving down and towards the right of the y-axis.

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Now, let s look at what happens to our graph when values smaller than four are used in the equation.

When values less than four were used, the curves in the graph became smaller. They moved closer to the y-axis.

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What do you think will happen when substitute 4 for 1?

Our graph produced an ellipse with the line y=x.

Try replacing the 1 with .5. Draw the line y=x in order to see how our graph will align.

We can infer that using values smaller than 1 in this equation will produce the same type of diagram from our first example. The only difference is the diagram lies along the y-axis as opposed to the x-axis.

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__More representations
using values <1__

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Thus far, we have been using positive values in substitution from the 4 in our original equation. Let s see what happens to our graph if we use negative values instead.

I constructed the line y=x, again to see how my diagram would align. It appears that the smaller our negative value becomes, the closer the curves approach the y-axis. Each of the curves intersect at the points (0, 1) and

(0,-1). The stems of the curves align closer to the y=x line when the negative values are greater.

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__Other Explorations:__

Let s examine the equation below:

This graph is very similar to the graph from our initial equation. Therefore, setting our initial equation equal to 1 does not make a big difference in the diagram that is produced.

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Try using other values for z:

Notice that as we increase the values of z, our curve in the second quadrant begins to level out. Even in the case where z=3, the curve becomes smaller, producing a closed circular figure. The curves in Quadrant IV, however, become increasingly bigger as z becomes larger.

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**Based on what we have
discovered, what equation will the following graph produce?**

**If you guessed the answer
below, you are absolutely correct!**