In this investigation we examine several functions in which x and y are raised to the same degree, which varies, and their sum is equal to one. In algebraic form we would state the function this way:
We are most familar with this function as it exists for a circle. That is, the graph of is a circle with a radius of one.
The problems starts off by asking us to graph this unit circle, and then to graph successive functions, increasing the exponent (a) by one each time. We continue to set the equation equal to one for consistency, but actually, it becomes clear very shortly that the same phenomena we are about to discuss will continue to hold true regardless of the value of the equation.
After we graph the functions in which a = 2,3,4 and 5, a definite pattern immediately comes into focus.
When the exponent is even, the graph is symmetrical about both the x and the y axis, and it "comes back around on itself". The perfect example of this, of course, is the case in which the exponent is 2 - that of the circle. We know the circle to be the most symmetrical of all two-dimensional objects. Notice, though, how the graph is similar in shape when the exponent is increased to 4. Similar, but different, in the sense that the shape expands outward along the diagonal axes, looking more squarish.
What about the functions in which the exponent is odd? Not only do they not come fully around the origin and they increase and decrease infinitely, but they are not symmetrical about either axis. In fact, the domain does not even exist for cases in which both x and y are negative (in quadrant III).
These characteristic shapes are reminiscent of the graphs of quadratic and cubic functions. While quadratic and all functions raised to an even power are symmetrical about the y axis, and have at least one minimum or maximum point, functions raised to an odd power increase and decrease without bound. They also contain one or more points in which the inflection of the graph changes. (The figure below displays typical quadratic and cubic functions.) This seems to be the case in our functions as well. We will evaluate these inflection points later in this investigation.
Our investigation naturally leads us to functions with much larger exponents. Will we continue to see the same pattern? Let's graph some of these and see.
Yes, the pattern does hold true. But now let's examine further. Since it seems that we are dealing with two distinct classes of functions, we'll focus on each one separately to see what mysteries they behold.
The functions with odd exponents are the most interesting to me, probably because of their asymmetrical shape. Why are they shaped this way? Why do they curve and turn where they do?
Let's begin by listing some significant coordinate points for functions with both large and small odd exponents, as we look at these turning or inflection points.
|x^3 + y^3 =1||x^5 + y^5 =1||x^25 + y^25 =1||x^101 + y^101 =1|
|inflection pt. A||-1||1.26||-1||1.15||-1||1.03||-1||1.0075|
|inflection pt. B||0.7937||0.7937||0.87055||0.87055||0.97265||0.97265||0.99316||0.99316|
|inflection pt. C||1.26||-1||1.15||-1||1.03||-1||1.0075||-1|
What do we notice? Well, they all go through the points (1,0) and (0,1). The left-most inflection point (call it A) is always at x= -1, and the right-most inflection point (call it C) is at y = -1. The corresponding x and y values at these turning points starts off greater than one, but then decreases towards one as the exponent increases. We wonder if it will ever reach one. That is, is there an exponent large enough such that the point (1,1) would be on its graph? We suspect not.
Now to the really interesting inflection point in the middle, B. Why does each graph completely change directions here? What is significant about this point? I wanted to know what its coordinates were, and I tried to pin it down by stopping at the point on each graph that "looked" like it was the turning point. The coordinates for point B increased steadily as the exponent increased, again moving towards one, but never quite reaching it. OK, this seemed appropriate and definitely matched with the graph. But the x and y values were oh so close, yet not the same as each other. I accepted it, but it didn't seem reasonable.
OK, now its time for a little algebra to really put things into focus. Our spreadsheet list the significant turning points, but why is this so? We'll use to start our explanation of what this function is really saying.
We know that the curve represents all the points in which . As the first term increases, the second term decreases, and vice-versa, such that their sum is always equal to one. When x is smaller than -1, y is corresondingly large. As x increases, y must decrease. But why are the values exactly what they are? Well, let's see.
(-2)^3 + (2.08)^3 = 1 or (-8) + 9 =1
(-1)^3 + (1.26)3 = 1 or (-1) + 2 = 1
(0)^3 + (1)^3 = 1 (here's the point (0,1))
(1)^3 + (0)^3 = 1 (here's the point (1,0))
(2)^3 + (-1.93)^3 = 1 or 8 + (-7) = 1
In each case, the sum of x and y cubed is equal to one. And what of inflection point B? This is the point in which x and y are exactly equal to each other. That is, each term is equal to .5, for a sum of one. This is the exact 'center' point, so to speak. It's the point in which the magitude of the first and second terms changes place.
In our example, this point is (.7937)^3 + (.7937)^3 = 1.
When the exponent is 5 it's (.87055)^5 + (.87055)^5 = 1.
And when it's 25, the point is (.97265)^25 + (.97265)^25 = 1.
At an exponent of 101, (.99316)^101 + (.99316)^101 = 1.
And so my initial discomfort with the fact that I couldn't pin down values for which x and y were equal to each other at Point B was correct. In fact, this is the point in which the values are exactly equal. It was my instrument that was not fine enough.
Notice how the x and y values of this central inflection point keeps increasing towards one. Does it ever reach one? The answer to this is the same as the answer raised by one of the questions above. Can (1,1) ever be a solution to this equation? Basic arithmetic tells us that 1 + 1 can never equal 1. And detailed inspection of the graphs, even at the largest exponent values, bears this out. The graph is carefully carved out at this point.
One last question for now. Why are there no solutions to the equation in quadrant III? The answer, as always, is found in the algebra. The values for x and y cannot be negative at the same time if the equation is to hold true. Take the challenge, see if you can find simultaneous negative solutions for x and y.
For further investigation, I'd like to know why the x and y values never dip below x = y? Is x = y an asymptote, or is it included in the graph? The answer is on the tip of my brain, but not quite there.
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