### Melissa Silverman

The equation

describes a circle with a radius of 1 and a center of (0,0).

If x and y are raised to an even power, such as

then the graph is an enclosed shape.  Notice that as x and y are raised to a higher (even) power, the enclosed shape becomes more and more similar to a square. Like the graph above, the square shape has a center of (0,0).

If x and y are raised to an odd power, such as

,

then the graph has a similar square-like quality around the origin and approaches the line y=-x as x and y are increasing.

As x and y are raised to a higher odd power, the graph gets closer and closer to a square corner around the origin and approaches y=-x as the values of x and y increase.

So if x and y are raised to an even power, then the graph will be an enclosed shape, centered around the origin, that approaches a square.  As x and y become larger, the graph becomes more and more square. If x and y are raised to an odd power, then the graph will have a curved shape that approaches a corner about the origin and approaches y=-x as x and y become larger.

Suppose x has a coefficient, such as in the equation

The width of the circle has decreased to form an ellipse. The center of the ellipse remains at (0,0). The height of the ellipse remains at 2 units.

As the coefficient increases, the ellipse becomes more rectangular, while the height and the center of the shape are unaffected. Inversely, as the coefficient becomes smaller, the ellipse broadens and becomes wider.

Changing the coefficient of y changes the height of the enclosed figure. As the coefficient decreases, the shape resembles an ellipse. As the coefficient increases, the figure becomes more like a rectangle.

This investigation helps visualize the impact made on a simple equation by altering the components of the equation. By changing and isolating different components of an equation, conclusions can be drawn about the functions of the components. The graph of the equation takes on many forms, from enclosed figures to large hyperbolas. By altering the equation's characteristics, one can conclude upon the role of each part of the equation and gain a deeper understanding of the equation.

In an educational setting, this process can help students understand equations. By being able to change equations and graph them instantly, students can see the impact that their changes are making. Changing coefficients, exponents, and other elements in an equation correspond to different characteristics when graphed. Having control over those changes can help students comprehend equations and their uses. In a classroom, students could conduct a similar investigation to the one above. Once students have learned the basic formula for a circle, they may be encouraged to generate a list of questions. What happens when I add a coefficient to the x or y term? How can I move the center of the circle so it is not at the origin? What if the equation is equal to something other than one? I think this sort of questioning, followed by computer time to find answers to these questions, would be a great educational experience for the students. Much like in science class, they can ask questions, form hypotheses, and then test them using technology. And because each student would be questioning different things, the activity could be concluded by letting the students share their discoveries with the class. This would encourage mathematic communication and foster curiosity because students would want to know what everyone else was discovering, in addition to being eager to share their own findings.

This process could easily be applied to all levels of mathematics. For early algebra students, this same process of discovery could be applied to the equation of a line or parabola. For advanced geometry students, this could be conducted for any of the conic sections. Geometry students can test area and volume formulas. This is a general educational concept that can be applied sucessfully to many mathematical concepts.