1.

First, we graph

and see the following:

Let's vary the constant in the part of this equation and see what happens.

Now, let's look at

Looking at the three images above, it may be hard to notice a difference in the graphs with varying constants. To see the change more clearly, we can place all the equations on one graph as seen below.

Blue:

Magenta:

Red:

You can probably predict what will happen with the equation

Check your prediction here.

Now, what will happen with this equation: ?

What we see is quite different than the previous graphs and is a bit of a surprise.

Why do we have such an unusual graph? The image above could be interpreted as many different things, but mathematically we have an ellipse and a line. This can be shown by looking at the equation in a different light.

We start with our equation:

Now, simplify:

In the final equation above, we have the equation of a line being multiplied by the equation of an ellipse. See this concept graphically below.

Magenta:

Red:

For further exploration, let's try replacing our varying constant with a decimal and a negative number.

Blue:

Green:

Red: (Replacing the constant with -3)

We see above, that the graph changes drastically with both a negative number and a decimal number in the equation.

What happens if we vary the constant associated with the y side of the equation?

What happens if a constant is added to one side of the equation?

Blue: (Original Equation)

Magenta:

Red:

Green:

Light Blue:

Something interesting happens when the constant added or subtracted is 3.

Blue: (Original Equation)

Magenta:

Red:

It appears that the and equations are reflections of one another across the x-axis and the y-axis.

Let's look at a combination of the previous two graphs and see what happens with adding constants less than and greater than 3.

What will the graph become when we place it in 3-planes?

Now, we can see the maximum and minimum points from our 2-D graphs.

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