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 it’s gonna involve the rates of change. Now the rate of change is actually equal to the change in 𝑦 over the change in 𝑥, or 𝑑𝑦 over 𝑑𝑥, so what were gonna have to look at is actually look at how our 𝑥 values change and how our 𝑦 values changed to enable us to find our rates of change.
If we look at our 𝑥 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 𝑦 values, well if we go from 32 down to 25, that’s 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 I’m gonna do is I’m gonna have a look at the rate of change for the first two values. Of the first two values, the change in 𝑦 is negative seven cause it decreases by seven; the rate of change in 𝑥 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 𝑥 values are always increasing by three and our 𝑦 values are always decreasing by seven.
So a change in 𝑦 divided by change in 𝑥 will always be the same, so therefore let’s 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 it’s 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.