Linear and exponential growth — Basic example
About this video
A table of x and y values is tested against four candidate equations. The instructor checks each point, eliminates choices that fail, and then notes that the y-values grow by doubling, so the relationship is not linear and must be an exponential form that increases with x.
You'll be able to
- test candidate equations against points from a table.
- recognize linear versus exponential growth from how y changes with x.
- explain why a decaying exponential cannot match increasing y values.
Key words
- Linear relationship
- A relationship where y changes by the same amount each time x increases by one.
- Exponential relationship
- A relationship where y is multiplied by a fixed factor as x increases by one.
- Substitution
- Plugging a table's x value into a candidate equation and checking whether it gives the table's y value.
- Decay
- A pattern where a base between 0 and 1 raised to larger powers gives smaller values.
Transcript
- [Instructor] We're asked which of the following equations relates y to x for the values in the table above? Pause this video and see if you can have a go at that. All right, so we could just go point by point or coordinate by coordinate, depending how you wanna think about this.
So, first of all, when x is equal to one, y is equal to zero. So is that true right over here? When x is equal to one, this is gonna be 1/2 plus one, which is not equal to zero. That would be three halves, so we can rule this one out. When x is equal to one, let's see, two times one is two, minus two is equal to zero.
So this one is looking good so far, just based on that first x-y pair. And let's see when x is equal to one here, 1/2 to the first power is 1/2, times two is one, not zero. So we can rule this one out. All right, when x is equal to one here, this is two to the first power minus two, that's two minus two, that is equal to zero.
So this is looking good as well. We've already been able to remove two choices. Now, let's see, when we have two and two, let's just go to this one right over here. Two times two is four, minus two is equal to two. So this one's looking good on that second x-y pair.
Now, let's see, two to the second power is four, minus two, that is still, that is also equal to two. So this one's looking good. Now, let's think about three, six. So two times three is equal to six minus two, which would be equal to four, not six. So we could rule this one out.
And hopefully this will work for this one over here. So let's see, two to the third power is equal to eight minus two is indeed equal to six. So we like this choice right over here. Another way that you might have approached it is you could have said, "Look, this is not a linear relationship."
Because every time we're increasing x by one, here we're increasing by two, here we're increasing by four, here we are increasing by eight, here we're increasing by 16. If it was a linear relationship, every time we increased x by one, we would increase y by the same amount every time.
And so we know it's not linear, so you could rule out A and B. And you can also see that as x is increasing, y is increasing. So you know it would not be this exponential right over here, because here, as x increases, 1/2 to larger and larger powers would become a smaller and smaller value, y would decrease.
So that would also allow you to deduce that D is the choice.
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