About this video
The video graphs y = -f(x) for an exponential function with a base of one third, which becomes 1 minus (1/3) to the x. It checks candidate graphs two ways: by substituting easy x-values such as 1 and -1 to get specific points, and by reasoning about the end behavior as x becomes very negative or very large. Both methods point to the graph that passes through (1, 2/3) and (-1, -2) and flattens toward y = 1.
You'll be able to
- rewrite y = -f(x) by distributing the negative sign across the function.
- check a candidate graph by substituting easy x-values.
- use end behavior to confirm which graph matches a transformed exponential.
Key words
- Reflection over the x-axis
- Replacing y with -y, which turns y = f(x) into y = -f(x) and flips the graph vertically.
- Exponential function
- A function where the variable is in the exponent, such as (1/3) to the x, whose values grow or shrink quickly.
- Negative exponent
- A power such as (1/3) to the negative one, which equals the reciprocal, here 3.
- End behavior
- How a function's values behave as x becomes very large or very negative.
- Horizontal asymptote
- A horizontal line the graph approaches but does not reach as x grows large.
Transcript
- [Instructor] We're told the function f is defined by the equation above, right over here. Which of the following is the graph of y is equal to negative f of x in the x, y plane? Pause this video and see if you can work through that. All right, now let's work through this together.
So first of all, let's think about what y is equal to negative f of x is equal to, well that's going to be the negative of the right-hand side over here. So this is going to be, we could write it as negative 1/3 to the x. And then if we distribute the negative, it'll then be plus one, or we could even write it like this.
This is the same thing as one, minus 1/3 to the x. Now there's two ways that you could approach it. You could think about the behavior of this expression as x becomes very large or x becomes very small. Another possibility is we try a table with some easy values.
So let me start actually with the table method right over here. So let me just try some easy xs to calculate and figure out what the corresponding y is when y is equal to negative f of x. So one easy one would be, well, we could try zero, but we could see every one of these graphs have the point zero, zero so that won't really help us differentiate between the graphs that well.
But let's try the point one and let's try the point negative one. And those are useful because they're pretty easy to calculate. So let's see, when x is equal to one, 1/3 to the first power is just 1/3. One minus 1/3 is 2/3. Now when x is equal to negative one, what's 1/3 to the negative one power?
Well, that is equal to three. And so one minus three is equal to negative, negative two. Now let's see, just by inspection, if we can tell which of these graphs contain these two coordinates. So one, 2/3. So one, 2/3. That looks about right. So this one is looking pretty good.
B, one, 2/3. This is definitely larger than 2/3, so I would rule that one out. One, 2/3, this looks like negative 2/3. So I'd rule C out. One, 2/3. Definitely nothing right over here, so I'd to rule D out. So A is already looking pretty good, but let's just confirm negative one, negative two.
Negative one looks like the point. Negative two is there as well. So I'm feeling very good about A. And you could even check negative one, negative two, the graph doesn't go through there. Negative one, negative two. Doesn't go through there. Negative one, negative two does not go through there.
So we really like choice A. Another way that you could have thought about it is when x is very negative, when x is very negative, 1/3 to a very negative power is the same thing as three to a very large power. And so you'll have one minus three to a very large power.
Well, three to a very large power is going to be a large value. So one minus, that's gonna be a very negative value. So when x is very negative, your graph should get very negative. And this one checks that box. As x gets more and more negative, this graph is just really getting more and more negative.
And then as x gets larger and larger and larger, more and more positive. Well, 1/3 to larger, larger positive exponents is just going to get closer and closer and closer to zero. So you're going to approach one as x gets larger and larger. And it looks like that is what is happening with this graph right over here.
As x gets larger and larger and larger, y is approaching one. So once again, we like choice A.
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