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# Recognizing linear functions

Video transcript

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.