yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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) ).

More Articles

View All
TRAVEL CATACLYSM! (Wackygamer Commercial/Machinima)
Planning a trip to Azeroth this winter? You should! Recent events have caused a dollar to zero and gold exchange rate to skyrocket. Book now! Here are four helpful tips that will make your visit easy and painless. While property is always a risky investm…
High Seas Rivalry | Wicked Tuna: Outer Banks
I’m stuck. We’re staying. Pretty sure Fren’s even staying. Yeah, he has to, though; his title’s on the line. Yeah, he knows. He hasn’t said a word on the radio to us. Uh, he probably won’t. We got three fish; Frenzy’s got four. I got to admit it, I absol…
Why Vertical LLM Agents Are The New $1 Billion SaaS Opportunities
This is their first ever experience talking to this Godlike feeling, you know, AI that was all of a sudden doing these tasks that would take me, when I practice, like a whole day. And it’s being done in a minute and a half. The whole company, all 120 of u…
Sketching exponentials - examples
Now we’re going to take the ideas from the last video and learn how to sketch in these exponentials really rapidly. Now I want to move this up, and we’ll do some a couple of examples. Here’s an example circuit I’ve already set up. It’s an RC circuit. Thi…
How people actually end up buying a corporate jet from us.
Anybody come in there and just be like, “Hey dude, I saw the thing,” and end up buying a plane? Yes, the answer is definitely yes. Not only have I had people just walk in the showroom and say, “I’m looking to buy something,” they sign an agreement right t…
Follow Mexico's 'Bat Man' on a Search for Vampire Bats | Short Film Showcase
[Music] To an untrained eye, you see a rainforest, but someone who has a little bit of information of what was going on there can see the effects of humans all over the place. [Music] The Maya lived here for over 1,500 years, sustaining densities that wer…