Determining the effects on f(x) = x (multiple transformations) | Algebra 1 (TX TEKS) | Khan Academy

We're told here is a graph of a segment of f of x is equal to x. That's this here, and then they say h of x is equal to 1/3 * f of x minus 5. Graph h. So think about how you would approach this before we do this together.

All right, now I'm going to do this step by step. So before doing all of h of x, first I just want to think about what would 1/3 of f of x look like. Then we're going to think about what happens if we then were to subtract five.

1/3 of f of x, whatever f of x is for a given x, it's just going to be one third of that. So when x is equal to three, instead of f of f of three being three, 1/3 of that would be one. Similarly, f of 0 is 0; 1/3 of that is zero. f of -3 is -3; 1/3 of that is -1.

So what I'm drawing here—what I'm drawing here—and I could put those big endpoints so it looks like a segment. Right over here, this is the graph of 1/3 f of x.

Now, if we want to do 1/3 of f of x minus 5, what we need to now do is shift this graph down by five. So whatever 1/3 f of x is, it's now going to be five less than that. So if we take this point and we shift down one, two, three, four, five, we go down here.

Actually, we scroll down a little bit, and if we were to take this point and we were to shift down one, two, three, four, five—let me scroll down a little bit more so that you could see that. If you shift this point down five, you come right over here.

Now we would shift—actually, do this in another color—it would look like this. So this right over here is the graph of h of x. This is 1/3 f of x minus 5.

Let's do another example. So let's take this step by step, and what I am going to do here is make a little bit of a table. You will eventually realize that this just shifts the graph over. But if we take an x over here, and then we take x minus 3, and then we evaluate what f of x minus 3 is going to be.

Let's say let's do this at x equal 6, 3, and 0, and you might realize in a second why I'm doing those points. When x is 6, x minus 3 is 3, and then f of x minus 3 is going to be the same thing as f of 3, which is equal to 3.

So for this part right over here, if I just want to graph f of x minus 3 when x equals 6, it is equal to 3. By the same logic, when x is 3, x minus 3 is 0; f of x minus 3 is going to be the same thing as f of 0, which is equal to 0.

So when x is equal to 3, f of x minus 3 is equal to 0. You can see that when you subtract a number here from within the function, we're not subtracting it from the function; we're subtracting it from x before it's input into the function.

It's actually shifting us to the right by three. To verify that, we could try when x is 0; x minus 3 is -3, and so f of x minus 3 is the same thing as f of -3, which is equal to -3.

So when x is equal to 0, f of x minus 3 is equal to -3. So it would look like this—just this part. f of x minus 3. Let me write that. That's f of x minus 3; it shifted us to the right by three.

Now let's just think about what that is if we were to multiply it by -2. Well, there we're just going to scale all of these values by -2. So when x is 6, if we were getting to 3 before, well, you multiply that by -2; you're now going to be at -6.

So instead of that point, we're now going to be at -6. Let's take this point over here—if we were at -3 before and now we are multiplying by -2, we are now going to be at positive 6. So it's going to look like this.

So this right over here is the graph of that segment of -2 * f of x minus 3.

Now let's finish all this up. Let's do the full g of x and add the four. Well, we're just going to shift every point up by four. So this point is going to go from 6 to 10; this point is going to go from -6—shifting it up by four is going to go to -2.

So we are going to have a segment that looks like this, and we're done. This right over here is g of x is equal to -2 * f of x minus 3 + 4, and we're done.