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Graphing proportional relationships from an equation

Let's graph the equation y = 2.5x. For every increase of 1 in x, y increases by 2.5. We call this the "unit rate" or "slope". The graph shows a proportional relationship because y changes at a constant rate as x changes and because y is 0 when x is 0. Created by Sal Khan.

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Video transcript

We're asked to graph y is equal to 2.5 times x. So we really just have to think about two points that satisfy this equation here, and the most obvious one is what happens when x equals 0. When x equals 0, 2.5 times 0 is going to be 0. So when x is 0, y is going to be equal to 0. And then let's just pick another x that will give us a y that is a whole number. So if x increases by 1, y is going to increase by 2.5. It's going to go right over there, and I could graph it just like that. And we see just by what I just said that the unit rate of change of y with respect to x is 2.5. A unit increase in x, an increase of 1 and x, results in a 2.5 increase in y. You see that right over here. x goes from 0 to 1, and y goes from 0 to 2.5. But let's increase x by another 1, and then y is going to increase by 2.5 again to get to 5. Or you could say, hey, look, if x is equal to 2, 2.5 times 2 is equal to 5. So this is a legitimate graph for this equation, but then they also tell us to select the statements that are true. So the first one is the equation does not represent a proportional relationship. Well, this is a proportional relationship. A proportional relationship is one where, first of all, if you have zero x's, you're going to have zero y's, where y is equal to some constant times x. And here, y is equal to 2.5 times x. So this is definitely a proportional relationship, so I'm not going to check that. The unit rate of the relationship is 2/5. So I'm assuming-- this is a little ambiguous the way they stated it. I'm assuming they're saying the unit rate of change of y with respect to x. And the unit rate of change of y with respect to x is, when x increases 1, y changed 2.5. So here they're saying when x changes by 1, y changes by 0.4, 2/5 is the same thing as 0.4. This should be 5/2. 5/2 would be 2.4. So this isn't right as well. The slope of the line is 2.5. Well, this looks right. Slope is change in y over change in x. When x changes 1, y changes 2.5. So change in y, 2.5, over change in x, 1. 2.5 over 1 is 2.5. And you could also see it looking at the form of this equation. y is equal to-- this is the slope times x. So that's right. A change of 5 units in x results in a change of 2 units in y. Well, let's test that idea. We know when x is 0, y is 0. So if x goes from 0 to 5, what's going to happen to y? Well, y is going to be 2.5 times 5. 2.5 times 5 is 12.5. So y would not just change 2. It actually would change 12.5. So this isn't right. A change of 2 units in x results in a change of 5 units in y. Well, we see that. A change in 2 units of x results in a change of 5 units in y. That's exactly what we graphed right over here. These two points show that. So this is definitely true.