How Long to Jump Rope?

To begin my grandma decided to graph the quadratic ax^{2} + bx + c = y when a = 1, b = 0, and c = 0. This will be her parent function or the function that we will be comparing all the other graphs to in order to see how the graph is changing with different coefficients (More on Parent Functions). With that in mind my grandmother used her graphing calculator to see the graph of her beginning quadratic.

Grandma's Parent Function y = 1x^{2} + 0x + 0

Now that we know what the graph will look like with only one minute of jump roping accomplished (and no other workout) she wanted to know how the graph would change if she jump roped for longer period of time. She came up with the following five graphs where the coefficient a is how many minutes my Grandma jump roped this week.

2x^{2} + 0x + 0 |
4x^{2} + 0x + 0 |
6x^{2} + 0x + 0 |
12x^{2} + 0x + 0 |
20x^{2} + 0x + 0 |
---|---|---|---|---|

This demonstration of graphs showed that when a is increasing the graph seems to be getting closer and closer to the y-axis. My grandma described this as stretching the graph. My Grandma wanted to keep her experiment going by working out for longer and longer periods of time seeing if the graph would continue in stretching in this way. However the following week my Grandma was struck with a virus and was unable to jump rope for long periods of time, although she never willing to stop anything she started still jump roped just not for very long. She was able to come up with five graphs were the coefficient a is how many minutes my Grandma jump roped the next week.

1/2x2 + 0x + 0 | 1/4x2+0x+0 | 1/8x2+0x+0 | 1/12x2+0x+0 | 1/20x2+0x+0 |
---|---|---|---|---|

After seeing these graphs with Grandma, let me know that what was happening was we were "shrinking" the graph. When the coeffecient was between 0 and 1 the graph was shrinking, but when a was greater than 1 the graph was shrinking. To show that this was true for all numbers inbetween her selected coefficients a, I had to use a special program. I had been using the program Graphing Calculator for a while and I was able to create some movies that showed my Grandma how the graphs were changing when a > 1 and when 0 < a < 1.

With this my Grandma decided to stop Jump Roping for a while and decided to follow with her doctors other activities.