Determining the effects on f(x) = x (multiple transformations)
About this video
The video graphs combinations of transformations of a segment of f(x) = x, step by step. First h(x) = 1/3 f(x) - 5 (compress, then shift down 5); then g(x) = -2f(x - 3) + 4 by shifting right 3, multiplying by -2, and shifting up 4.
You'll be able to
- graph a function with several transformations by doing one step at a time.
- use a table of x, x - 3 and f(x - 3) to see that f(x - 3) shifts the graph right by 3.
- track how specific points move through each transformation.
Key words
- transformation
- A change to a graph such as a shift, a flip, a stretch or a compression.
- horizontal shift
- Moving a graph left or right; f(x - 3) moves it 3 to the right.
- vertical shift
- Moving a graph up or down by adding or subtracting outside the function.
- scale
- Multiply every output by the same number, which stretches, squashes or flips the graph.
Transcript
- [Instructor] We're told here is a graph of a segment of f of x is equal to x. That's this year. And then they say h of x is equal to 1/3 times f of x minus five, graph h. So think about how you would approach this before we do this together. All right, now, I'm gonna do this step by step.
So before doing all of h of x, first, I just wanna think about what would one 1/3 of f of x look like? And then we're gonna think about what happens if we then were to subtract five. So 1/3 of f of x, whatever f of x is for a given x, it's just gonna be 1/3 of that.
So when x is equal to three, instead of it, f of three is three, 1/3 of that would be one. Similarly, f of 00, 1/3 of that is zero. f of negative three is negative three, 1/3 of that is negative one. So 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 wanna do 1/3 of f of x minus five, what we need to now do is, shift this red graph down by five. So whatever 1/3 f of x is, it's now gonna 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, let me 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, you get there. If you shift at this point down five, you come right over here.
And now, we would shift, actually, lemme 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 five. 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'll eventually realize that this just shifts the graph over. But if we take an x over here and then we take an x and minus three and then we evaluate what f of x minus three is going to be. Let's say, let's do this at x equals six, three, and zero. And you might realize in a second why I'm doing those points.
When x is six, x minus three is three, and then f of x minus three is going to be the same thing as f of three, which is equal to three. So for this part right over here, if I just want to graph f of x minus three, when x equals six, it is equal to three. By the same logic, when x is three, x minus three is zero, three minus three.
And so f of x minus three is gonna be the same thing as f of zero, which is equal zero. So when x is equal to three, f of x minus three is equal to zero. 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 and to verify that, we could try when x is zero, x minus three is negative three. And so f of x minus three is the same thing as f of negative three, which is equal to negative three. So when x is equal to zero, f of x minus three is equal to negative three.
So it would look like this, just this part, f of x minus three. Lemme write that. That's f of x minus three. It shifted us to the right by three. Now, let's just think about what that is if we were to multiply it by negative two. Well, there we're just gonna scale all of these values by negative two.
So when x is six, if we were getting to three before, well, you multiply that by negative two, you're now going to be at negative six. So instead of that point, we're now going to be at negative six. And let's take this point over here. If we were at negative three before, and now we are multiplying by negative two, we're now going to be at positive six.
So it's going to look like this. So this right over here is the graph of that segment of negative two times f of x minus three. Now, let's finish all this up. Let's do the full g of x and add the four. Well that's, we're just going to shift every point up by four.
So this point is gonna go from six to 10. This point is going to go from negative six, shifting it up by four, it's going to go to negative two. And 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 negative two times f of x minus three, plus four, and we're done.
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