Video Transcript
Determine whether the table below represents a linear or nonlinear function.
Before we can actually determine whether the table below represents a linear or nonlinear function, we actually have to see what linear and nonlinear actually mean.
For a function to be linear, it would mean that it has to have a constant rate of change. And for a function to be nonlinear, it would actually have to have a variable rate of change. Okay great, so now we actually know what linear and nonlinear mean. We can actually start to solve this problem.
And looking at our definitions for both, we know that itโs gonna involve the rates of change. Now the rate of change is actually equal to the change in ๐ฆ over the change in ๐ฅ, or ๐๐ฆ over ๐๐ฅ, so what were gonna have to look at is actually look at how our ๐ฅ values change and how our ๐ฆ values changed to enable us to find our rates of change.
If we look at our ๐ฅ values, we could see that actually to get from three to six, we add on three; six to nine, we add on three; and from nine to 12, we add three. Taking a look at our ๐ฆ values, well if we go from 32 down to 25, thatโs subtracting seven; 25 to 18, also subtracting seven; and 18 down to 11 is subtracting seven.
Now if we were to look at the rate of change, so what Iโm gonna do is Iโm gonna have a look at the rate of change for the first two values. Of the first two values, the change in ๐ฆ is negative seven cause it decreases by seven; the rate of change in ๐ฅ is actually three because actually increases by three each time.
So therefore, our rate of change would be negative seven over three. But this rate of change is negative seven over three is actually gonna be the same throughout this table, because our ๐ฅ values are always increasing by three and our ๐ฆ values are always decreasing by seven.
So a change in ๐ฆ divided by change in ๐ฅ will always be the same, so therefore letโs answer the question. Well if we take a look at our definitions, the definition that suits us here is the top one that says that itโs gonna be a linear function because there is a constant rate of change, so therefore we can deduce that the table represents a linear function.
