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Graphing shifted functions | Mathematics III | High School Math | Khan Academy

2m read
·Nov 11, 2024

We're told the graph of the function ( f(x) = x^2 ) we see it right over here in gray is shown in the grid below. Graph the function ( G(x) = (x - 2)^2 - 4 ) in the interactive graph, and this is from the shifting functions exercise on Khan Academy.

We can see we can change the graph of ( G(x) ), but let's see, we want to graph it properly. So, let's see how they relate. Well, let's think about a few things. Let's first just make ( G(x) ) completely overlap. Well, actually, that's completely easier to say than to do. Okay, there you go. Now they're completely overlapping, and let's see how they're different.

Well, ( G(x) ) if you look at what's going on here, instead of having an ( x^2 ), we have an ( (x - 2)^2 ). So, one way to think about it is when ( x = 0 ), you have ( 0^2 = 0 ); but how do you get zero here? Well, ( x ) has got to be equal to 2. ( (2 - 2)^2 = 0^2 ) if we don't look at the -4 just yet.

So, we would want to shift this graph over two to the right. This is essentially how much we shift to the right. It's sometimes a little bit counterintuitive that we have a negative there, because you might say, "Well, negative? That makes me think that I want to shift to the left." But you just have to remind yourself, "Okay, for the original graph, when it was just ( x^2 ), to get to ( 0^2 ), I just have to put ( x = 0 ). Now, to get a ( 0^2 ), I have to put in a 2." So this is actually shifting the graph to the right.

And so, what do we do with this -4? Well, this is a little bit more intuitive, or at least for me when I first learned it. This literally will just shift the graph down. Whatever your value is of ( (x - 2)^2 ), it's going to shift it down by four.

So, what we want to do is just shift both of these points down by four. So, this is going to go from the coordinate ( (5, 9) ) to ( (5, 5) ), and it's going to go from ( (2, 0) ) to ( (2, -4) ). Did I do that right? I think that's right.

Essentially, what we have going on is ( G(x) ) is ( f(x) ) shifted two to the right and four down—two to the right and four down. Notice if you look at the vertex here, we shifted two to the right and four down, and I shifted this one also. This one also, I shifted two to the right and four down.

And there you have it. We have graphed ( G(x) ), which is a shifted version of ( f(x) ).

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