For Assignment 1, I chose to investigate item 1. The problem is to examine what happens when "4" in the following equation is replaced with other values.
This equation's graph looks like this...
Let's see what happens when we change the equation by substituting numbers from 2 through 7(inclusive) in the place of 4.
Now we'll see what happens if we use the negative values of the same numbers.
The above represents the curve of the greatest magnitude.
The above represents the curve of the smallest magnitude.
We conclude that positive numbers > or = 2 create predictable curves of increasing magnitude and that the roots that lie on the x-axis change while the roots on the y-axis remain the same.
The above two equations represent the extreme values of this graph. The other values range between -2 and -7 as replacements for the original value of 4. We conclude that values between -2 and -7 would create curves that wrap more closely around the y-axis. Fairly predictable stuff and not very exciting.
So far we have avoided looking at a graph for values 0 and 1. In mathematics, we sometimes see special things happening with zero. So let's see what happens in a graph with values of -2, -1, 0, 1, 2. (See below.) Now that's different! Did zero create the red lines? Look at the equations to the right of the graph. The red figure was created by 1.
The red equation creates an ellipse centered at the origin with a line that bisects it. Since all the other equations generated continuous curves, this shape was unexpected. Let's look first at the red line. It passes through the origin (0,0) and through (.5,.5), (1,1),(2,2),(3,3).... It also passes through (-1,-1),(-2,-2),(-3,-3)... The red line tells us that every number pair (x,y) is a solution to the red equation when x=y. What does the ellipse tell us? That one's harder. I sat down with a calculator and an Excel spreadsheet and tried to "guess-timate" values that satisfied the equation. I wasn't very successful. So I let Graphing Calculator show me some number pairs not on the axes that were solutions. Here's some examples: (0.56414932,-1.1546007), (0.825,0.30430106), (-0.1875,1.0804783), (-0.8125,-0.30430106), (0.13968862, 0.92281138). I also discovered that creating a graph with values that get increasingly closer to 1 create curves that get increasingly closer to the line/ellipse figure. The numbers 1, 1.095, 1.1, 1.15, 1.18, 1.19, 1.195 created the graph below. So I found a number pair on the curve where the original 4 was replaced by 1.095. That pair is (0.13565639,0.91690639). What connection exists between an ordered pair from the line/ellipse and an ordered pair from a curve that closely approximates it?
We've looked at values outside of the range from -2 to 2. Let's see what happens is we use numbers between 2 and 1 and -2 and -1. If we create a graph to illustrate the values -1, -1.1, -1.3, -1.5, -1.7, -1.9, Graph 4 looks very similar to Graph 3 above. Similarly, if we use 1.1,1.2,1.3,1.4,1.6,1.7, we create Graph 5, which looks similar to Graph 2, but with more extreme curves.
Graph 1. From the graph, we can see that roots (solutions) of the equation are (0,0), (0,1), (0,-1), (2,0), and (-2,0). Every point on the line is also a solution to the equation but we can only estimate those number pairs from the graph.
As we vary the equation with values closer and closer to one, the curve draws closer and closer to the line/ellipse figure and the ordered pairs that were solutions to the red equation become closer and closer approximations to solutions for the curve. Compare (0.13565639, 0.91690639) from the curve to (0.13968862, 0.92281138) from the line/ellipse. Plugging these values into each of the equations develops solutions that are within .0000005 of each other. That's pretty darn close.
So what have we learned?
- Changing a single value in an equation may drastically change the representation (graph) of the equation.
- Changing a single value in an equation may minimally change the representation (graph) of the equation.
- Sometimes the change is expected and sometimes it's a surprise.
- Graphs(equations) that appear to be different may have solutions that are infinitesimally close to each other.
Here's some more fun.... Modify the equation as seen below and then vary the constant from 0 to 5 and see what happens....
Assignment 1:Examining Curves
By Rebecca L. Adcock