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Graphing shifted functions

Given the graph of f(x)=x², Sal graphs g(x)=(x-2)²-4, which is the graph of f shifted 2 units to the right and 4 units down.

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Video transcript

- [Voiceover] We’re told, "the graph of the function f of x "is equal to x-squared." We see it right over here in grey. It’s shown in the grid below. "Graph the function g of x is equal to "x minus two-squared, minus four "in the interactive graph." This is from the shifting functions exercise on Khan Academy, and we can see we can change the graph of g of 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 of x completely overlap. Okay, there you go. Now they're completely overlapping. And let’s see how they’re different. Well, g of x, if you look at what's going on here, instead of having an x-squared, we have an x minus two-squared. So, one way to think about it is, when x is zero, you have zero-squared is equal to zero. But how do you get zero here? Well, x has got to be equal to two. Two minus two-squared is zero-squared, if we don’t look at the negative four just yet. And so, we would want to shift this graph over two to the right. This is essentially how much do 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 have to remind yourself is like, well okay, for the original graph, when it was just x-squared, to get the zero-squared, I just have to put x equals zero. Now to get a zero-squared, I have to put in a two. So this is actually shifting the graph to the right. And so, what do we do with this negative four? 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 minus two-squared, it's gonna shift it down by four. So what we wanna do is just shift both of these points down by four. So this is gonna go from nine, and this is gonna go from the coordinate five comma nine, to five comma, if we go down four, five comma five. And this is gonna go from two comma zero, to two comma negative four. Two comma negative four. Did I do that right? I think that’s right. What, essentially, what we have going on is, g of x is f of x shifted two to the right and four down. Two to the right and four down. And 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 of x, which is a shifted version of f of x.