PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
One important concept in the study of polynomials is the reciprocal transformation. What happens when we take the reciprocal transformation of a function, or one over the function Specifically, there are ways to create the graph of the reciprocal transformation of a function from the graph of the function itself. The reciprocal transformation is important in the definition of rational functions.
I want to take a look at a particular transformation called the "Reciprocal Transformation" given the graph of a function y=f of x what does the graph of y equals 1 over f of x look like? To figure that out I want to start with a simple example. Let's use the graph of y equals 1 minus a half x to graph y equals 1 over 1 minus a half x.
First thing I want to do is graph y equals 1 minus a half x and that's pretty easy alright this is going to be a line with y intercept 1 and slope negative one half. So it's gong to go down 1 over 2 and right away when you have 2 points for a line you can graph the line immediately so let's graph it. So that's our line, now how do we get points for this graph 1 over that? Well you can just take reciprocals right? For example this point has a y coordinate of 1, the reciprocal of 1 is 1 so the reciprocal graph will pass through this point. This point has a y coordinate of a half the reciprocal of that is 2 so the reciprocal graph will pass through this point. Let's pick a nice integer here we're going to have a y coordinate of 2, so the reciprocal will have a y coordinate of a half. Here we have a y coordinate of 3, the reciprocal is one third and you can kind of see what's going to happen, as the graph goes up to infinity the reciprocal goes down to zero. And we get this kind of shape, now what happens past here notice that the y values of my lines are getting close to zero.
Well let's take a table of values, I've got x 1 minus 0.5x and 1 over 1 minus 0.5x. Let's let x get closer and closer to 2 and see what happens as we get close to this point. So we've already done x equals 1, let's do 1.5 now half of 1.5 is 0.75 and 1 minus that if 0.25, the reciprocal of 0.25 is 4 okay so it's going up. If we wanted to we could plot that point, it would be up here 1.9 half of that is 0.95 1 minus 0.95 is 0.05 and the reciprocal of that is 20. You might be able to tell already that as these numbers get closer to 2 these numbers are going to get closer to 0, and these numbers are going up to infinity. So that's what out graph is going to do, as we get close to 2 this graph is going to move up to infinity. Alright and that means that we have a vertical asymptote that x equals 2. So let me draw that in, it's a vertical line, a vertical line that the graph is going to get closer and closer to as it moves up.
Now what happens on this side, again we can just plot some points like on the line at this x value which looks like 4, we're going to have negative 1 the reciprocal of -1 is -1 so my reciprocal graph will actually go through that point. Here the y value is negative a half the reciprocal of that is negative 2 so we'll go through this point and let's say here at 3 we have 3 halves, the reciprocal of that is two thirds and you can see that we're going to get a similar graph, a similar kind of graph down here as up here. Now is this going to happen? As x approaches 2 from the right are we going down to negative infinity? Let's check really quickly with some values, 2.5 half of that is 1.25, 1 minus 1.25 is negative 0.25 and the reciprocal of that is negative 4 so this is 1 over 1 minus 0.5 x.
What about 2.1 half of that is 1.05, 1 minus that is minus 0.05 and the reciprocal of that is negative 20. So yeah you could see that as this goes to 2, we're approaching 2 from the right, this is approaching zero and this is going to negative infinity. And that verifies that the graph should actually just go down, straight down to negative infinity, so we've got a vertical asymptote x equals 2 and you might also recognize that the x axis is a horizontal asymptote] right the graph is getting closer and close to it as x goes to infinity or as x goes to the negative infinity.
So this purple graph is a graph of my reciprocal function, y equals 1 over 1 minus 0.5x and the red graph is the graph of my original line y equals 1 minus 0.5x.