Reflecting functions: examples | Transformations of functions | Algebra 2 | Khan Academy

What we're going to do in this video is do some practice examples of exercises on Khan Academy that deal with reflections of functions. So, this first one says this is the graph of function f. Fair enough. Function g is defined as g of x is equal to f of negative x. Also fair enough. What is the graph of g?

On Khan Academy, it's multiple choice, but I thought for the sake of this video it'd be fun to think about what g would look like without having any choices, just sketching it out. So pause this video and try to think about it, at least in your head.

All right, now let's work through this together. So we've already gone over that g of x is equal to f of negative x. So whatever the value of f is at a certain value, we would expect g to take on that value at the negative of that.

So, for example, we can see that f of 4 is equal to 2, so we would expect g of negative 4 to be equal to 2. Because once again, g of negative 4, we could write it over here: g of negative 4 is going to be equal to f of the negative of negative 4, which is equal to f of 4.

And so we could keep going with that. What would g of negative 2 be? Well, that would be the same thing as f of 2, which is 0. So it would be right over there. What would g of 0 be? Well, that would be the same thing as f of 0 because the negative 0 is 0. And f of 0 is right over there; it looks like negative 2.

And so you can already see where this is going. And we've already talked about it in previous videos that if you replace your x with a negative x, you're essentially reflecting over the y-axis. So g is going to look something like this. It is going to look something like this. Once again, g of negative 6 would be the same thing as f of 6. And so that would be the graph of g.

If you're doing this on Khan Academy, you'd pick the choice that looks like this; that would give a reflection over the y-axis. Let's do another example. So here, once again, this is the graph of the function f, and then they say what is the graph of g.

So pause this video, at least try to sketch it in your mind what g should look like. All right, so in this situation, they didn't replace the x with negative x in f of x. Instead, g of x is equal to the negative of all of f x. In fact, we could rewrite g of x like this: we could say that g of x is equal to—notice all of this right over here—that was our definition of f of x.

So g of x is equal to the negative of f of x. So, instead of it being f of negative x, it's equal to the negative of f of x. So one way to think about it is we can see that f of 0 is 2, but g of 0 is going to be the negative of that, so it's going to be equal to negative 2.

And so you could keep going with that. You could see that whatever f of a certain value is, g of that value would be the negative of that. So it would be down here, and so g of x would be a reflection of f of x about the x-axis. So g of x is going to look something like something like that—a reflection about the x-axis.

And so once again, I'm kind of going to pick the choice that would actually look like that. Let's do another example; this is strangely fun. All right, so here we're told functions f (so that's in solid in this blue color) and g dash (so that's right over there) are graphed. What is the equation of g in terms of f?

So pause this video and try to think about it. So the key is to realize: how do we transform f of x? Actually, they labeled it over here; this is f x right over here. In order to get g, so f of negative x would be a reflection of f about the y axis.

And so it would intersect there; it would have this straight portion like this. And I'm just experimenting right now to the straight portion like this, and then it would go up. And let's see; f if f of negative x. So when you input 6 into it, that would be f of negative 6, which is 6. So it would go up there.

So f of negative x would look something something like this—something like that. So the purple is f of negative x. Now, that doesn't quite get us to g, but it gets a little bit closer. Because it looks like if I were to take the reflection of f of negative x, f of negative x about the x-axis, it looks like I'm going to get to g.

And so how do you reflect something about the x-axis? Well, we saw it in the example just just now; you multiply the entire function by a negative. So we could say that g is equal to the negative of f of negative x, is equal to the negative of this.

So we're doing both reflections: we're flipping over the y-axis, and we're flipping over the x-axis to get to g. Let's do one more example. So once again, they've graphed f, they've graphed g, and they've said f is defined as this right over here. What is the equation of g?

So they're not just asking it in terms of f. They just want to know what is the equation of g. Pause this video and try to think about it. Well, you can see pretty clearly that this is a reflection across the y axis.

And a reflection across the y axis, you can see pretty clearly that g of x is equal to f of negative x—f of negative x. How do we know that? Well, whenever we take f of x if and we get that value g at the negative of that value takes on the same function value, I guess I could say.

Or another way to think about it is we could just pick this point: negative 8, f of negative 8 is equal to a little over 4, but g of 8 is equal to a little over 4, is equal to that same value. And so what is the equation of g?

Well, we just have to rewrite this so that we can write it out as an equation. And so we could write out g of x is equal to—if I were to replace all of the x's here with a negative x, what would I get? I would get 4 times the square root of 2 minus—instead of an x, I will have a negative x—and then the minus 8 is outside of the radical.

And so we would have g of x is equal to 4 times the square root of 2 plus x minus 8. And we're done.