Assignment 1: Exploring Graphs with a degree of greater than or equal to 2
We will be exploring many graphs of the following equation:
First letís graph this equation when n = 2.
When we graph this equation with n = 2, , we noticed that it looks like a circle. This makes sense because in mathematics, we have a unit circle that is a circle with a radius of 1. Thus since for all x, and since the reflection of any point on the unit circle about the x or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle not just those in the first quadrant.
Now letís see what happens when n = 3? Will it have a similar shape as the graph of when n = 2?
Wow, this graph is shaped completely different than when n = 2. This is becoming interesting. The graph maintains a similar shape around 0<x<1 but then curves out along the y = -x line. Before I visit this idea further, letís graph some more equations to see if they follow the trend of the first two graphs.
Letís see what the graph will look like when n = 4? Will it look more like n = 2 or n = 3? But then I will become even more curious to see what n = 5 looks like.
Letís view these four equations together graphed on the same Cartesian plane and see if we can notice a pattern and/or relationship.
What patterns are visible from these graphs?
We noticed that as n increases and is an even number, the graph looks like a circle. It starts as a circle but continues to stretch out as n increases. However, as n increases and is odd, the graph follows the trend of n = 3. As a result we see extended curves that donít close and then it wraps along line and is asymptotic to the y = -x line. Itís so interesting to see that when n is odd the graph almost becomes the y = -x line except near the region 0< x <1 it stretches and curves around that line.
We are guessing that if we continue in this manner, for even powers of the equation, we will get a shape between a circle and square and for the odd powers we will get a curve passing through (0, 1) and (1, 0). Also each curve is above the previous curve between x = 0 and x = 1, but below the curve for other values of x.
So, letís explore further. What do we expect to see when n = 24 and n = 25?
So as we observe these different graphs, we began to ask ourselves why do odd graphs during this specific interval have an asymptote of y = -x? We noticed that the graph will always be less than 1 but greater than 0 because you are always subtracting the independent variable from 1; consequently the value of y (dependent variable) will always be less than 1.
So letís test this assumption. Consider the point (3, -3). Letís replace the x value with 3 and the y with -3. We notice that it does not satisfy the equation anywhere because the y will be less than 1.
Explore the animation to the right to see if the pattern holds true for larger values of n.
Notice that if n is not an integer what happens to the graph.
In conclusion, we can discover more intriguing results by changing the values of n.
Last but definitely not least, letís look at what happens when we graph the following equation: