Comparing linear functions
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f is a linear function whose table of values is shown below. And they give us three different x-values and the corresponding f of x values. Which graphs show functions that are increasing faster than f? So when we're talking about increasing faster, we're really talking about a higher rate of change of y with respect to f, or a higher rate of change of the vertical axis with respect to the horizontal axis, which is another way of saying which of these have a steeper slope than the function f? So let's see what the change in our vertical axis is with respect to our change in our horizontal axis. Once again, the Greek letter-- this triangle is the Greek letter delta, which is shorthand for "change in." So this is the change in f over the change in x. So we see over here, when x changes by 1, the value of our function changes by positive 5. And it's linear, so that's true. Between any two points, the ratio between our change in f and our change in x is the same. If we go up 1 again, we have plus 1 in the x-direction, we are once again increasing by 5. If you start from this point and go all the way here, so if you go plus 2 along x, you're going to go plus 10 along f. So it would be 10 over 2, which is still 5. So either way, the slope, or the rate of change of the vertical axis with respect to the horizontal axis, is 5 for f. Now, let's see which of these increase faster. Well, a isn't even increasing. So A is decreasing. As x increases, y is decreasing. So that definitely can't be the case. If we look at this one right over here, it looks like-- let's see, if we start over here, if we increase 1 along the x-direction, if our change in x is 1, it looks like our change in y is exactly 5-- 1, 2, 3, 4, 5. So it looks like for choice B, our slope is exactly 5, or our change in y over change in x is exactly 5. So it's not increasing faster than f. It's increasing the same as f. Now let's look at C. So I'm going to try to find a point where it looks like I have an integer point right over here. So that's the point, negative 3, negative 3. And if I move 1 in the x-direction, it looks like I'm increasing by more than 5. I'm increasing 1, 2, 3, 4, 5, 6, 7, 8, it looks like. So this one looks like it has a slope of 8. So this one is increasing faster than f, so we'll circle that right over there. And now let's look at this choice. So if we start right over here-- and I just picked this point because that's at a nice integer coordinate. It's at the point 2, negative 4. If we increase x by 1, then we increase y by 1, 2, 3-- looks like about 3 and 1/2, definitely not 5. In order for it to increase as fast as f, it would have to increase by 5, so it would have to be up here. So it would have to go 1, 2, 3, 4, 5. It would have had to have been up here. The line would've looked something more like that just to match f, much less grow faster than f. So D does not meet the criteria. It is only C.