for michelle lewis only

X	Y
2.0	0.500000
4.0	0.250000
6.0	0.166667
8.0	0.125000
10.0	0.100000
12.0	0.083333
14.0	0.071429
16.0	0.062500
18.0	0.055556
20.0	0.050000

Question 1.

When talking about an inverse variation, the concept being brought up entails the expression of two variables in an equation from which their product results to a specific constant value therein. In narrowing down our scope, we can opt to express the same using distinct variables; x, y and k in the equation below. The whole equation can be given in a number of different forms, but primarily, x*y should equal a constant k. [xy = k] where k is a constant not equal to zero. Let us take a pair of our values and equate within the aforementioned equation. Where y is 0.5 and x is 2, the constant obtained is a distinct 1. This is a similar case for the figures 0.25 and 4. The notable characteristic we can construe from the equation and the nature of the values within the graph dictates that as x increases from 2 to 4 progressively, y on the other hand reduces from 0.5 to 0.25. This either works with a concept of doubling the figures or halving the others.

In inverse variation, y varies inversely as x. This means that when x increases, y will decrease significantly with the same factor. The expression xy is a constant obtained by multiplying the distinct x and y co-ordinates respectively. This characteristic is very evident in our scatterplot graph and all the values plotted to correspond with our general equation. The trend for such an equation as y=1/x seems to progress distinctly and as expected within such a rational function. Over time, the function and the graph bring forth another distinct feature as the graph tends to deviate away to an infinite direction following the y axis of our chart. When x is near zero, y is seen to grow instantaneously, this type of behavior is always referred to as blowing up. In other words, the function blows up at the point where x = 0.

Question 2.

Within our rational function, there are quite a number of behaviors associated with such a given equation. All we need is to first define is the direction within which the same equation originates from and what side of the planes of the axes that it lies upon. It would be considered correct and prudent to note that as we approach the point 0 from the right within the x-axis, values calibrated within the y-axis are seen to start increasing progressively. Posing a question regarding the limit of 1 divided by x as x approaches 0 would be a parallel concept describing our problem altogether. Within the equation, the sequence dictates that as x becomes a significantly small number, the other sequence of y results in y becoming a large number due to reciprocation. Such a condition within the graph where our x details tend to zero whereas those of y increase in magnitude, can be associated with an approach towards positive immunity. At this point, the graph does not have a likelihood of coming into contact with the y-axis as the same runs parallel to an infinite figure. This is also allocated a positive sign as it moves in the positive direction as figures in the y plane. When we talk about a vertical asymptote, the input of values do approach from a given point and as a result, sees a magnitude increase gradually without bound.

We need to note down that in a rational function; at its vertical asymptote, the graph equation is said to be undefined. Some people would opt to term such a situation as the position where the value of x equals 0. From the data of my equation and scatterplot, we could also hold that the trend still corresponds with the theoretical provisions considered to be documented. Significantly, as we approach 0 from the right, the curve moves to an infinite direction.

Question 3.

After evaluating our equation y=1/x, we have come up with quite a number of values that we have used to come up with the scatterplot in our case. We noted significantly the inverse variation of the variables while our k remained constant. When the value of x was seen to move up, a similar idea was observed in the y section but in a negated manner. As the values scaled down, the trend that seemed normal and uniforms for quite a good dimension starts to change gradually. In other words, the proportionality issue seems to raise the eyebrows more so when we are just a few points from point 0. This however seems to continue and shift as the graph of our equation moves to 0. In trying to fit a model to the curve originating from the plotting, we opt to use the polynomial function concept. The polynomial curve seems to back in the upward direction alongside the vertical asymptote. The model for our graph can be said to be fit to some extent by virtue that at some point, there is uniformity and cogence just before the drastic twist that fosters a curve towards an infinite direction in a positive manner. This vertical asymptote of ours raises a rational function perspective therein at the point of convergence.

y = 1 / x

2 4 6 8 10 12 14 16 18 20 0.5 0.25 0.16666700000000001 0.125 0.1 8.3333000000000004E-2 7.1429000000000006E-2 6.25E-2 5.5556000000000022E-2 0.05