What happens when we start with a quadratic function in the form of

and very **a** from negative numbers, to zero and positive numbers?
Most students will recognize the equation, when a is set equal to zero as
the equation of a circle with a center at (0,0) and a radius of 3. This
is the red circle in the graph below, to the left. The center graph has

with two values of **a,** both 0 and -1. The graph to the left has
values of **a** of 0 and +1.

From examining the graphs with **a** equaling +1 and -1, we observe that
they appear to be ellipses. By trying fractional values of **a** from
-2 to 2, we can see more evidence that supports this observation. The larger
the absolute value of **a**, the flatter the ellipse becomes. The following
graph has been enlarged to make it easier to see the changes to the graph
when a changes. The values of a range from O for the red circle to 1/2,
1, and 3/2 for the green ellipse.

When we graph the equation with values of

How do we know that these are straight lines? By examining

with a =2, we notice that it can be factored as (x+y)(x+y)=9. If we take
the square root of both sides, the result is that x+ y = 3 or x + y = -3.
Both of these are equations of a line, and are in fact the lines that were
graphed.

When the absolute value of a is increased to be greater than 2, a fourth type of graph appears, the graph of a hyperbola. As

The graph below shows the effects of changing a from values of 0 for the
red circle to 3 to 9 to 29 for the brown hyperbola.

The following two graphs show how the graphs change when

The last graph simply shows all the graphs, for seven different values of

A question that remains, and will be left for you to consider, is what
happens to the graphs if a different constant other than nine is used in
the original equation. Does the constant have to be a square number? Does
it have to be an integer? Does it have to be positive?

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