Graphing rational function transformations
At about 68s the narrator says the shifted horizontal asymptote of y = f(x) - 4 is 'y equals four'; it should be y = -4. A correcting note was added at 69.5s and the questions use y = -4.
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Starting from the graph of f(x) = 1/x, the video shows that y = f(x) - 4 shifts the whole graph, including its horizontal asymptote, down 4 units. It then works through y = f(-1/2 x) point by point and shows it flips the graph over the y-axis and stretches it horizontally.
You'll be able to
- describe how adding or subtracting a constant to f(x) shifts a rational function's graph and its horizontal asymptote.
- find points on y = f(bx) by evaluating f at b times x.
- explain how a negative b flips the graph over the y-axis and how |b| < 1 stretches it while |b| > 1 compresses it horizontally.
Key words
- asymptote
- A line the graph gets closer and closer to but never touches.
- vertical shift
- Moving every point of a graph up or down by the same amount, caused by adding or subtracting a constant to f(x).
- reflection over the y-axis
- Flipping a graph left-to-right across the y-axis, which happens when the input x is replaced by -x.
- horizontal stretch
- Pulling a graph away from the y-axis, which happens in f(bx) when the magnitude of b is less than 1.
- horizontal compression
- Squeezing a graph toward the y-axis, which happens in f(bx) when the magnitude of b is greater than 1.
Transcript
- [Tutor] What we have depicted here is the graph of Y is equal to f of x, where f of x is equal to one over x. What we're going to do in this video is explore what happens if we were to transform f of x a little bit. For example, what if we wanted to plot, I'll do this in a new color.
What if we wanted to plot Y is equal to f of x minus four. What would that look like? Well, what this would do if you add or subtract a constant to the value of f of x, you're just gonna be shifting it up or down in the vertical direction. Since it's minus four here, we're just gonna shift everything down by four.
So this point right over here goes down 1, 2, 3, 4. Another way to think about it. So we have our asymptote, it's looking something like this. The one way to think about even what would be at zero zero, which we have both vertical and horizontal asymptotes there, that's also shifted down four.
So you have your asymptotes, your horizontal asymptote, instead of it being at y equals zero, now it's at y equals four. And so if I were to draw it, we're getting closer and closer into it but never quite touching it. And then on the left hand side, we are also, we are also shifted down like something like that.
So notice everything just shifted down by four. Now let's do another example. I'll do a different type of transformation in purple. Let's imagine Y is equal to F of negative one half x. Now this always messes with my brain a little bit. I always like to do a little bit of a mental model of, okay, if I input a given value of x, what happens to it?
And let's just imagine a little bit. So if I have x, what is f of negative one half x? Or maybe, maybe I should, well, yeah, let's write it this way. x, then what is negative one, half x. And then we could think about what is f of negative one half x? So if we were to say, take the 0.1, you take negative, which if you just took f of one, that puts you at this point right over there.
But negative one half of one, that's negative one half. So then negative f of negative one half of one is f of negative one half. So instead of it being equal to one, it's what would have been equal in our original function, in our original function up here at x equals negative one half.
So at x equals negative one half, our original function was negative two, and so that's what we're going to be right over here. We're now going to be at negative two when we were at one before. And we could try that with a few other values. So for example, when x equals two, negative one half of two is negative one.
So this is f of negative one. f of negative one we know is negative one. So at two we are at negative, we are at negative one. So you already might be seeing something interesting that's happening here. And this can lead to a general principle. One, it looks like we have flipped over the Y-axis.
And indeed when you have some constant here, and if that constant is, has a negative sign in front of it, you are going to be flipping over the x axis. If you just, or sorry, over the Y axis. If you just had an f of negative x, that would be a pure flip over the Y axis, over this line right over here.
But when we're multiplying it by, when this constant has a magnitude less than one, you're also stretching it out. So you're flipping and stretching it out. And so that's why it looks like this. Not only it's flipped and stretched out a little bit, and it's going to be that on both sides, it's gonna be flipped.
And I know I'm not drawing this probably perfectly. It's gonna be flipped and stretched out. So if this value right over here, if the magnitude of it is less than one, you're gonna be stretching it out horizontally. If it's greater, the magnitude is greater than one, you're going to be compressing it horizontally.
And if you have a negative number here, then you are also going to be flipping it over the Y axis.
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