Finding the zero of a linear function (IMRA 2026 publisher-initiated change)

0:00- [Instructor] We're told to graph the line f of x is equal to negative one half x plus three and then figure out the zero of this function. So if you have some paper handy, try to do it yourself or at least think about how you would approach it. Okay, now let's do it together.

0:14So first let's graph this line. Let me get some graph paper here. There you go. And let me see. All right, so let me make some x and y axes here. So let me say that this is the x-axis. Whoops, you can't see that. Let's say that this is the x-axis right over there.

0:34That is the x-axis. And then let me make, let me maybe make this the y-axis like that. So that is the y-axis. And then this is, I'll just make every square equal to on. So that is one, two, three, four, five and so on. And then this is one, two, three, four, five and so on.

0:59Let me just mark them all off. Hopefully you get the idea here and I'll mark the numbers that seem relevant. So if we want to graph the line f of x is equal to negative one half x plus three, the way I would think about it is, well, first of all, what is the y intercept?

1:15The y intercept is the value of f of x. We're going to graph y is equal to f of x. So the first thing you want to think about is what is y equal to when x is zero? Or essentially what is f of zero? So when x is zero, so let me just write this right over here. F of zero is equal to negative one half times zero plus three.

1:40Well, that's just going to be three. So we're going to have the point x equals zero, y equals three, or the y intercept is three. And you might recognize this is in slope intercept form, that what I am circling in blue is the slope and what I am circling in yellow is the y intercept.

2:00So the point zero three is going to be on this line. And now we could either look for another point on the line to plot it or we could just think about its slope. Its slope is negative one half. So that means that every time you move forward x by one, you're going to go down.

2:18You're going to go negative one half. Or another way to think about it, if you move x forward or if you increase x by two, you're going to decrease y by negative one half times that, which is one. So you go forward by two, you go down by one. So that would be a point on our line.

2:38That would be a point on our line and that would be a point on our line. And so the line would look like, let me get a little tool out for this. All right. Let me put that point there. I want to do this neatly. And then let me put this point right over there. Let me make sure. Okay, that's pretty good.

3:00And then let me just graph the line. There you go. Looks pretty nice. So I've graphed the line. This is y is equal to f of x right over there. And then they say, what is the zero of this function? So the zero of the function is the x value that makes the function equal to zero.

3:23So another way to think about it is what x value will make f of x equal to zero? Or another way to think about it is you could say negative one half x plus three is equal to zero and then solve for x. You can visually see it here that it looks like it happens at x equals one, two, three, four, five, six.

3:43This right over here is x equals six or this is a 0.6 comma zero. But let's just make sure that that's the case. We could solve for x here. So if we subtract three from both sides, we get negative one half x is equal to negative three. And then if you multiply both sides times the reciprocal of this coefficient, so by negative two by negative two, that cancels with that.

4:05We're just left with x on the left hand side. And on the right hand side, this is equal to six. Or another way to think about it is if you put x equals six into this original expression right over here, negative one half times six is negative three plus three is indeed equal to zero.

4:23So what is the zero of this function? You would say it's six or you would say x equals six is the zero.