Basic Transformations - Problem 1

A transformation occurs when we move the graph, in some direction so for this particular example we are going to look at the graph of x² minus 2. We know what the graph of x² looks like. It looks like a parabola.

So we are going to make a little table and see what happens to our values as we look at the graph x² minus 2. So some of pens out of the way, plug in -2, -2² is 4 minus 2 is 2. Plug in -1, -1² is 1, minus 2 is -1, plug in 0 goes away so we are just left with -2. Plug in 1, 1² minus 2 is again -1, 2² minus 2 is 2 okay. So we plugged in our x values we found our f(x) factor so y values. So what I want to do now is compare our y values here with the y values for our basic x² graph. So when we plugged in -2 before we ended up with 4 now we get 2. We plugged in -1 we had 1 now we end up with -1, 0, -2 and so on. If you notice the new value that came out is just two less than the old value. So what this graph actually does is just takes our original graph and moves it down 2.

It takes any single point and moves it down 2 so where it was at the origin and now down here, (1,1) down here (–1,1) is over here, everything just gets shifted down 2, not the exact graph but basically we took our graph moved everything down 2.

Let’s take a look at what we think happens if we deal with f(x) is equal to x² plus 3. We could make a table but the same exact thing is going to happen instead of subtracting 2 from every value, we are just going to add 3, all that’s going to end up doing it moving the graph up three units. So what this graph is going to be is before our bottom, what we call the vertex is at the origin this is actually now going to be at three units, depending on how precise your teacher wants you to be they may want you to plot the rest of the points, in general I did some considered with the general case or the general shape so I know that I need to have my lowest point here and then the basic shape is going to stay the same look something like this.

So anytime you have a graph and then you are adding or subtracting subtraction moves it down adding moves it up.