Assignment 2
Jonathan Beal -
Perplexing Parabolas
In this exercise, I am going to explore and play with quadratic equations.
Specifically, y = ax^2+bx+c.
I can graph parabolas of different shapes and sizes by changing the values of the coefficients.
First I want to look at the terms of the function individually.
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For the third term for the equation, ax^2+bX+c, I am going to let c=1 to demonstrate how the function moves in a positive direction up the y-axis. | |||||
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Now I want to explore what manipulating each term does to the function and how this affects the other variables.
First I will look at examples of the first term.
Without the first term, the function devolves to a line going through 1 at each axis. | |
Look and see, when the coefficient in the first term is 1, then the graph is centered at .5 and it has no roots. | |
Now, we begin to see the characteristics of the value of the first term. The area between the legs of the parabola are coming closer together | |
And the parabola has closed up a lot with a big increase in the value of the coefficient. It almost has formed a line on top of the y-axis. |
And to turn the parabola upside down, the leading coefficient needs to be negative. To turn the parabola to the side, this requires a different function: ay^2+by+c=x |
Next, I will focus on the properties of the coefficient of the second term.
The coefficient is 0 in this function as a control example. Without the second term influencing the equation, the parabola is centered on the y-axis. The trough of the function is at 1. | ||||||
In this example, which we have seen a couple of times now, the parabola moves to the left by .5. | ||||||
I can see that the values of the second term stimulate the parabola to extend downward. | ||||||
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And now we are going to play with the third term in the equation. In ax^2+bx+c, c is the constant. It should not have any effect on the function other than the position on the graph.
When c = 0, the graph has roots at -1 and 0. This variable will make the function move up and down. Watch and be amazed! |
See. The function has moved up when c=1. The rest of the function has not changed. It is not wider or more narrow. Will this continue as the number get larger? |
In this equation, c=1 and the coefficient is 2. The function moved upward on the y-axis as it did when previously. This trend will continue as c gets larger and larger. We could expect then that if the numbers were negative that the graph would decrease in altitude as they got smaller and smaller.
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As I expected, the parabola's minimum or trough sunk when c = -1. |
Let's see what happens if we explore all variable movements at the same time.
First I show the negative effects of each term on the function.
Next, I show the movement of all the variables. We can see all the previous steps by each variable all at once and how each respective variable compares with another variables