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Graphing the Transformation y = a f(x) + k - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

There are many different transformations, and graphing transformations is different depending on what type it is. One of the most basic types of transformations is y = af(x) + b. Graphing transformations of this type involves creating a table of the original and the transformed function and graphing the second from the first. We should also recognize horizontal shifts, reflections, and horizontal compression.

We're talking about graphing transformations. Let's start with an easy transformation. y equals a times f of x plus k. Here's an example y equals negative one half times the absolute value of x plus 3.
Now first, you and I ide- identify what parent graph is being transformed and here it's the function f of x equals the absolute value of x. And so it helps to remember what the shape of that graph is. Absolute value looks like this, it's got a little corner at the bottom. And there are three key points that I usually like to start with. There's the point -1 1, the point 0 0 and the point 1 1, and my technique here is basically to take points of my parent graph and transform those points first and then plot the transformed points. So these are going to these are points of x absolute value x. And this point is really important. This is my this is my vertex right? The turning point of the graph and so I want to see where the vertex ends up because that'll be my new vertex.
Now, when you look at this function, the function's basically saying, multiply the absolute value of x by negative one half and then add three. Now this means two things. First of all, all the transformations are going to happen on this side of the column and secondly there what are the transformations? This multiplication by negative one half what does that do?
Well, first of all, multiplying by a negative number is going to flip the graph across the x axis. Multiplying by a half is a vertical compression of the graph, and adding three will shift the graph up.
So we have x, negative one half absolute value of x plus three. But you'll see all that when you do the Arithmetic on these numbers. Now first of all we'll just carry the x values over cause nothing's happening inside the absolute value. So those are the x values. And then for each of these absolute value of x values, I'm going to multiply them by negative one half and add three. So one times negative one half is negative one half plus 3 is 2.5, 0 times negative one half is 0 plus 3 is 3. And 1 times negative one half is negative one half plus 3 is 2.5.
That's not a bad start. Let's plot these points. We've got 0 3, this is 2, 3. We've got -1 2.5, so that's -1 is here 2.5 is here and one 2.5. Those points are kind of close to my y intercept. Let's just plot some points that are further out. So let's say oh -6 and 6. -6 the absolute value is 6, 6 the absolute value is 6. Those x values will translate right over oops positive 6 but what happens to the absolute value? Well we multiply by negative one half and add 3. So this times negative one half is -3 plus 3, 0. Again 6 times negative one half is -3 plus 3 is 0. So we get we get -6 0 and 6 0, and those are these points here and here and that's going to give us a much better graph if we use points that are further away from the y intercept. So this is going to be our graph. And you can see that the graph is reflected across the x axis right?
Normally, the absolute value graph opens upward like this. Now this one's opening downward. You can also see the vertical compression right? The slope is it's less steep than absolute value usually is and you can also see that the vertex has lifted up three units so you could see that vertical shift. This is our final graph.

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