Interpreting piecewise linear functions
About this video
Using a piecewise linear graph of water in a tank over time, Sal finds the interval where the amount changes fastest by comparing slopes. From 0 to 4 minutes the rate is 5 liters per minute, from 4 to 8 it is zero, and from 8 to 12 it is 8 liters per minute, which is the fastest; the decreasing part changes more slowly.
You'll be able to
- find the rate of change on each piece of a piecewise linear graph.
- compare rates using the steepness (absolute value of slope).
- explain why a decreasing piece can still be compared by its size of change.
Key words
- piecewise linear function
- A graph made of straight line pieces joined together, each with its own slope.
- rate of change
- How much the output changes for each unit of input, like liters per minute.
- absolute value
- The size of a number without its sign, so a steep fall and a steep rise can be compared.
Transcript
- [Instructor] The graph shows the amount of water in the tank in liters as a function of time in minutes, all right? During which time interval is the amount of water changing the fastest? Pause this video, have a go at that. Okay, so the key is, is the interval where the water is changing the fastest is going to be wherever we see the highest, or I guess you could say the lowest slope where maybe the absolute value of the slope is actually the highest.
So, let's see, when we go between zero and four, looks like our water changed 20 liters over four minutes, so that's the slope right over here. That is 20 liters over, let me write it this way. 20 liters over four minutes, which is equal to five liters per minute.
It's definitely not this next interval. Here, we're not changing at all. So, this is from four to eight minutes. We could just rule that one out. Now, this next interval is an interesting one, because here, we're going from, it looks like roughly eight minutes to, or from eight minutes to 12 minutes, so once again, this is a four-minute change that is happening here.
And yeah, write minutes here so you don't think I'm writing meters. And then let's see what the change in liters is. The change in liters over that same time is we went from 20 to 52, so that's 32 liters. So, 32 liters, 32 liters in four minutes is equal to eight liters per minute.
So, that's looking pretty good. It's definitely higher than choice A, so we could actually rule that one out. I'm liking this one. And now this next interval, it's decreasing, but is it decreasing at a faster rate than the interval we just saw? Well, we can clearly see that it's not, because if you wait another four minutes, we didn't get back to where we were before.
We definitely decreased at a reasonable rate, but if we decreased at the same rate, we would've gotten down to here. If we decreased at even a higher rate, we would've gone down even more, so we could rule this one out as well. And I like choice C.
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