Assignment 2 Writeup 8. Produce several (5 to 10) graphs of y= (x-d)^2 -2 on the same axes using different values for d. Does varying dchange the shape of the graph? the position?
The above-described function represents a parabola. Click here to read more on the history of the parabola.
At first, let's look at the parabola with values of d from 1 to 5. This means that all the values of d are positive so let's see what effect that has on the graph of the parabola. Here are the different parabola functions with these values of d.
Now, let's look at the graphs of these functions to see how the different values of d affect graphs of the function.
First, look at the parabola in red of
. This parabola has its vertex at (1,-2). If the parabola was centered at (0,0), we would move it to the right one and down two to get it at the point (1,-2). The vertex of a parabola is where it is centered at or where it opens up. Each of the other functions is very similar. The increase in d just moves each parabola to the right one space. D correlates to the position of each vertex of each parabola along the x-axis (horizontal axis).
Now that we have looked at the transformation of the parabola for positive values of d from 1 to 5, let's look further at the parabola for negative values of d from -1 to -5. An initial insight would tell me that the negatives values of d would now move the graph to the left instead of the right. The two negatives would make all the graphs be y= (x+?)^2 - 2. Below are five parabola functions with this characteristic.
Here are the functions, now let's look at the graphs of these functions to see how the parabola changes.
My intuition was right. The negative values for d did cause the graph of the parabola to be moved to the left. The first parabola of
has a vertex at (-1,-2). Again, this is the point where the parabola is centered and where it opens up. The other parabola functions with d from -2 to -5 move one value along the x-axis to the left. For example, the blue parabola function has vertex at (-2,-2). For the five functions, the only value that changes is the x-value. The y-values stay the same at -2.
Finally, let's take a look at the graph of the parabola when d = 0 or
. This parabola has a vertex at (0,-2) since d = 0. Remember, the x-value is the only value that changes when d is changed. The y-values remains at -2.
A final investigation involves looking at the graph of the parabola for values of d that are less than 1. I picked both positive and negative values of 0.5 and 0.2.
As you can see, the positive values of d just move it along the x-axis to the right that amount and for negative values of d, the parabola moves along the x-axis to the left. The graphs of these parabolas are closer together because of the small values of d that I chose.
This concludes my investigation of the following:
Produce several (5 to 10) graphs of y= (x-d)^2 -2 on the same axes using different values for d. Does varying dchange the shape of the graph? the position?
The shape of the graph is not changed by the different values of d. The only thing that varies is the parabola's position along the x-axis.
