Determining the effects on f(x) = x when replaced by af(x) or f(bx)
At about 52s the narrator says 'f of three is negative six' when he means g(3) = -6. The interventions use g(3) = -6.
The check-ins use the correct version. The video is Khan Academy's, unchanged.About this video
Using a segment of f(x) = x, the video graphs g(x) = -2f(x) by multiplying outputs by -2, which stretches the graph vertically by 2 and flips it over the x-axis. It then graphs h(x) = f(1/3 x), showing that h(6) = f(2) and h(-9) = f(-3), so multiplying x by a fraction stretches the graph horizontally.
You'll be able to
- graph a·f(x) by multiplying each output by a.
- explain that a negative a also flips the graph over the x-axis.
- graph f(bx) and explain why 0 < b < 1 makes the graph wider.
Key words
- vertical stretch
- Multiplying every output by a number bigger than 1, making the graph taller or steeper.
- reflection over the x-axis
- Flipping a graph upside down, caused by multiplying outputs by a negative number.
- horizontal stretch
- Making a graph wider by multiplying the input x by a number between 0 and 1.
Transcript
- [Instructor] We're told here is a graph of a segment of f of x is equal to x. And so they've graphed that segment right over here. And then they tell us that g of x is equal to negative two times f of x, and they want us to graph g. So think about how you would approach that.
Now let's work on this together. So g of x, whatever x is, I would evaluate f of x, and then I would multiply that by negative two. So let's pick, say when we are at x equals three, f of three, it looks like that is three. In fact, we know that's three 'cause f of x is equal to x.
So f of three is three. But if we wanna figure out g of three, that's going to be f of three times negative two, which is three times negative two, which is negative six. So f of three is negative six. Now let's think about negative five over here. When x is equal to negative five, we see that f of negative five is negative five, which makes sense.
But if we were to take g of negative five, that's going to be negative two times f of negative five. So it's gonna be negative two times negative five, which is positive 10. So it would get us up here. So really what you see happening when we multiply f of x by negative two, well, if we multiplied it by two, we would just be scaling everything up by a factor of two, but then that negative then flips it over the x-axis to get what we see right over here.
Let's do another example, but it's gonna be a little different. Here, instead of multiplying times our f of x, we're multiplying the x by a number here. So here's a graph of the segment f of x is equal to x. We see that again. And now they've defined h of x as being equal to f of 1/3x.
So let's graph h. So one way to think about it is, I know what f of two is. F of two is equal to two. Now, for H, I could actually input six in here. I could figure out what h of six is. How do I know I can do that? Because H of six is going to be f of 1/3 of six.
Or another way of saying it, h of six is going to be the same thing as f of two. So h of six is the same thing as f of two, which is two. And then we could do that on the negative side. For example, we know that f is defined at negative three,. F of negative three is negative three.
Now, if we were to go three times that value and we were say, what is H of negative nine? H of negative nine, we could go over here. H of negative nine is equal to f of 1/3 times negative nine, or it's gonna be the same thing as f of negative three. F of negative three is negative three.
So h of negative nine is negative three. So notice now we are scaling, we're making it wider when we multiplied inside of our function or as we multiply x times a fraction. If we multiplied this times a value greater than one, then we would be squeezing it in the horizontal direction.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.