The Department of Mathematics and Science Education

tiffani c. knight

Okay, first let’s look at the graph of

Next, we’re going to replace the 4 from the equation above with 5, 3, 2, 1.1, 1, 0.9, and -3. ( I am including the original equation to observe the relationship between the equations to the original one). Let’s observe:

It appears that as the number replaced gets smaller, the curve/hump of the equation gets smaller and starts to form an oval shape…elliptical?

And when the actual number is replaced with a negative, the oval becomes narrower and is vertical, not sideways. Hmmmm.

And when replaced with 1, that graph creates a straight line through (0,0) and it’s the only one that does that.

The interesting thing about them all is that they intersect in three places (0,1), (0,-1), and the origin (0,0).

Let’s explore how the graph looks when the number replaced is restricted to numbers between 0 and 1.

Well, it looks like the graph, when the number is replaced by 1, creates the oval and the 0 sets the inner lining, if you will, and all other numbers between 0 and 1 fill up the space between.

Let’s explore how the graph looks when a constant is added to one side of the equation.

Well, it no longer makes that elliptical shape. It does still have a similar “hump” on the left side. And none of them cross at the three points that they all crossed at before: (0,0), (0,1), and (0, -1).

Next I graphed . It produced an interesting looking 3D graph. I don’t know if it was supposed to happen, but I couldn’t save it. Open up graphing calculator and give it a try.