Since h(-5) = -(-5)-2 = 3, our function demonstrates these translations.Ī function reflected about the y-axis and then shifted horizontally: Since our original function was f(x) = x, our new function should be f(-x-2) = -x-2. Let's use the function f(x) = x and take the point (3,3) on that function. To understand more clearly, we can take a point on a function as an example. Reflected about the y-axis: g(-x) = f(-x-2) And we're done.A translation of a function horizontally two units to the right and then reflected about the y-axis: And so we would have g of x is equal to four times the square root Instead of an x, I will have a negative x. If I were to replace all of the x's here with a negative x, what would I get? I would get four times the And so what is the equation of g? Well, we just have to rewrite this so that we can write

To a little over four, is equal to that same value. F of negative eight isĮqual to a little over four, but g of eight is equal Or another way to think about it is we could just pick this Of x and we get that value, g at the negative of that value takes on the same function How do we know that? Well, for whenever we take f

And a reflection across the y-axis, you can see pretty clearly, that g of x is equal to f of negative x. Well, you can see pretty clearly that this is a reflectionĪcross the y-axis. Is the equation of g? Pause this video and

What is the equation of g? So they're not justĪsking it in terms of f. Graphed f, they've graphed g, and they've said f is definedĪs this right over here. We're flipping over the y-axis, and we're flipping over So we could say that g is equal to the negative of f of negative x. You multiply the entireįunction by a negative. Something about the x-axis? Well, we saw it in the example just now. Negative x about the x-axis, it looks like I'm going to get to g. Were to take the reflection of f of negative x, f of Now, that doesn't quite get us to g, but it gets us a little bit closer 'cause it looks like if I So when you input six into it, that would be f of It'd have the straight portion like this. Here, in order to get g? So f of negative x would be a reflection of f about the y-axis. How do we transform f of x, actually, they've labeled it over here, this is f of x right over What is the equation of g in terms of f? So pause this video and We're told functions f, so that's in solid in this blue color, and g dashed, so that's right You'd pick the choice that would actually look like that. So g of x is going to look something like that, a reflection about the x-axis. And so g of x would beĪ reflection of f of x about the x-axis. You could see that whateverį of a certain value is, g of that value wouldīe the negative of that. So it's going to be equal to negative two. So one way to think about it is we can see that f of zero is two, but g of zero is going toīe the negative of that. So instead of it being f of negative x, it's equal to the negative of f of x. X is equal to, notice, all of this right over here, that was our definition of f of x. All right, so in this situation, they didn't replace the x And then they say what is the graph of g? And so pause this video and at least try to sketch it out in your And if you're doing this on Khan Academy, you'd pick the choice So g is going to look something like this. And we've already talkedĪbout it in previous videos that if you replace your Same thing as f of zero 'cause a negative zero is zero. What would g of zero be? Well, that would be the Same thing as f of two, which is zero, so it What would g of negative two be? Well, that would be the G of negative four is going to be equal to f of the negative of negative four, which is equal to f of four.

Negative four to be equal to two because, once again, g of negative four, we could write it over here. That f of four is equal to two, so we would expect g of So whatever the value ofį is at a certain value, we would expect g to take on that value at the negative of that. Over that g of x is equal to f of negative x. Try to think about it, at least in your head. What g would look like without having any choices, What is the graph of g? And on Khan Academy, it's multiple choice, but I thought for the sake of this video, it'd be fun to think about Of exercises on Khan Academy that deal with reflections of functions. Going to do in this video is do some practice examples

Reflection over y axis