# Assignment 2: Problems and Explorations with Second Degree Equations

For this assignment we will explore the variations of the function y = ax^2. Where a is an element of real numbers. Below is the graph y = ax^2 where a varies from -10 to 10. Play the video to get an idea of how the graph changes. Please maximize your browser before continuing.

Now that we have a general idea of how the function changes as a changes let's take a closer look.

From the graphs above it is clear to see that the parabola y = x^2 becomes narrower as a increases. In the graph y = 10000x^2 the parabola looks as if it is almost a straight line along the positive x-axis.

Let's look at some final graphs before making some conclusions about the effects of the variable a on the function y = x^2.

After looking at the newest graphs above we see that when a is negative the parabola is reflected across the x-axis, and like the parabolas above the x-axis we see with those below the x-axis the larger the negative value of a the more narrow the parabola. Now the remaining question is why is the graph y = ax^2 behaving this way?

The function y = x^2 can be understood as the graph of the area of a square with side length x.

Unfortunately, many high school teachers do not thoroughly explain exactly why y = x^2 looks the way it does. Rather the "plug in numbers" strategy is used. When you trace the point (x, y) the green segment follows along the locus of the point (x, y) with reference to the point x. As stated above we know that the ratios of the similar triangles' side lengths show that the area of the square = x^2 = y = length of the green segment. Remember that the green segment is representing the length from (x, y) to (x, 0) and showing how it is equivalent to the area of the given square. When we multiply x^2 by a scalar the area of the square will change along with the length of the green segment, but our new point (x', y') will remain a part of the original locus. This does not fully explain why the graph behaves like it does, but it does help clarify what it means to say y = x^2.

For a quick reminder: According to Integrated Publishing (http://www.tpub.com/math2/10.htm), "The locus of an equation is a curve containing those points, and only those points, whose coordinates satisfy the equation."

To summarize the effects of the scalar a on the function y = x^2 we can recall what we observed above and tie up the loose strings. We can see in the area graph that the point (x', y') is above the point (x, y) for our point x was dilated by 2^(1/2) and y was dilated by 2. Our point (x, y) was stretched in the positive direction, which is essentially what happens when x^2 is multiplied by 2. By looking at the first two sets of graphs we can see how the graph of x^2 is stretched to the point of almost laying on top of the y-axis. The graphs are representing what happens when the points are dilated by all possibilities in the positive direction towards infinity. The parabolas flip across the x-axis when the scalar is negative. This makes sense because the function is now being stretched by a negative scalar in the negative direction towards infinity. When the function is multiplied by a scalar, a, the points that satisfy the equation y = ax^2 will be a dilation of the points that satisfy the original function y = x^2. The graph is being stretched in the positive direction or negative direction pulling the graph y = x^2 closer to the y-axis depending on the value of the scalar. And the closer the scalar is to zero the farther away the parabola will be from the y-axis.