Linear and nonlinear functions
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Deirdre is working with a function that contains the following points. These are the x values, these are y values. They ask us, is this function linear or non-linear? So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value. For example, for any one-step change in x, is the change in y always going to be 3? Is it always going to be 5? If it's always going to be the same value, you're dealing with a linear function. If for each change in x--so over here x is always changing by 1, so since x is always changing by 1, the change in y's have to always be the same. If they're not, then we're dealing with a non-linear function. We can actually show that plotting out. If the changes in x-- we're going by different values, if this went from 1 to 2 and then 2 to 4-- what you'd want to do, then, is divide the change in y by the change in x, and that should always be a constant. In fact, let me write that down. If something is linear, then the change in y over the change in x always constant. Now, in this example, the change in x's are always 1, right? We go from 1 to 2, 2 to 3, 3 to 4, 4 to 5. So in this example, the change in x is always going to be 1. So in order for this function to be linear, our change in y needs to be constant because we're just going to take that and divide it by 1. So let's see if our change in y is constant. When we go from 11 to 14, we go up by 3. When we go from 14 to 19, we go up by 5, so I already see that it is not constant. We didn't go up by 3 this time, we went up by 5. And here, we go up by 7. And here, we're going up by 9. So we're actually going up by increasing amounts, so we're definitely dealing with a non-linear function. And we can see that if we graph it out. So let me draw-- I'll do a rough graph here. So let me make that my vertical axis, my y-axis. And we go all the way up to 35. So I'll just do 10, 20, 30. Actually, I can it do a little bit more granularly than that. I could do 5, 10, 15, 20, 25, 30, and then 35. And then our values go 1 through 5. I'll do it on this axis right here. They're not obviously the exact same scale, so I'll do 1, 2, 3, 4, and 5. So let's plot these points. So the first point is 1, 11, when x is 1, y is 11. This is our x-axis. When x is 1, y is 11, that's right about there. When x is 2, y is 14, that's right about there. When x is 3, y is 19, right about there. When x is 4, y is 26, right about there. And then finally, when x is 5, y is 35, right up there. So you can immediately see that this is not tracing out a line. If this was a linear function, then all the points would be on a line that looks something like that. That's why it's called a linear function. In this case, it's not, it's non-linear. The rate of increase as x changes is going up.