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
Knowledge Makes the Existence of Resources Infinite
Knowledge is the thing that makes the existence of resources infinite. The creation of knowledge is unbounded. We’re just going to keep on creating more knowledge and thereby learning about more and different resources. There’s this wonderful parable of …
Inventing Graphics on Cave Walls | Origins: The Journey of Humankind
Early humans communicated with pictures and markings painted on cave walls and began to gradually work out symbols. As these markings spread and were understood and accepted, then you had the widespread transmission of ideas. We can see the very early day…
Ask Sal Anything! Homeroom Wednesday, July 15
Hi everyone! Welcome to our homeroom live stream. Sal here from Khan Academy. Uh, before we get into, uh, our guest, who is me today, it’s just going to be an ask me anything type of thing. So actually start putting your questions on Facebook and YouTube…
Fiscal and monetary policy in parallel | AP Macroeconomics | Khan Academy
In previous videos, we have talked at length about fiscal policy, and in other videos, we’ve talked at length about monetary policy. But now we’re going to talk about them together. Because at any given time in a country, there is some type of fiscal poli…
Flight at the Edge of the Ozone Layer | One Strange Rock
NARRATOR: 30 years ago, we discovered man-made chemicals had punched a hole in the ozone layer. Is that hole here to stay, waiting around to kill us? Today, we’re trying hard to find out. Morgan Sandercock is about to test an experimental plane, perfect f…
I FOUND THE 5 BEST BANK ACCOUNTS OF 2023
What’s up, guys? It’s Graham here. So the time has finally come: saving money is now officially profitable! That’s right; for the first time in my YouTube career, cash is once again King. If you have any amount of savings whatsoever, this applies to you, …