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Larousse Charlot

Graph Translation

Assuming that you are familiar with parabolas from trigonometry, the study of angles, I will go ahead and start with the graph of y = x^2.

Now, we should observe what happens to this parabola once a constant is added to the equation. For example, y = x^2 + d, where d is some integer. We then obtain

for d = 1

If you observe well, the graph is shifted up one unit on the y-axis. From this observation, we can conclude that the graph will shift down if our equation were to be y = x^2 - d for some integer d. Considering the possibilities, let us see what kind of shiftment that takes place with y = (x + d) ^2,

Notice that the graph has shifted to the left, on the x-axis, given that d = 1. A good question is "why to the left and not to right, as in the y-axis?" Well, the formula for such shifting is y = (x - d)^2. From there, the graph would shift to the right of the x-axis. However, we have y = (x + d)^2 which implies y = (x - (- d))^2. d is a negative integer.

If the equation y = (x - d)^2 - 2

were to present to you, can you infer what would the graph looks like or how is it shifted if it is a parabola? See the graph

As you can observe, the graph is shifted one on the x-axis and -2 on the y-axis.

Now, you probably have a sense of what's happening to the graph, we should observe several graphs simultaneously as the value of the constant d changes

As you can see there, the graph is moved only on the x-axis and its vertex is remained at -2 on the y-axis. So, as the value of d changes the graph, the x-cordinate of the vertex changes while the y-cordinate remains the same

To give a general idea of what's happening here, observe this movement of the graph.