Comparing linear functions: faster rate of change

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 is which of these have a higher 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 when X changes by 1 R the value of our function changes by positive v changes by positive 5 and it's linear so that's true of any between any two 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 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 + 2 if you go + 2 along X you're going to go + 10 + 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 definitely a isn't even increasing so we can say is decreasing as x increases Y is decreasing so this can't even be that definitely can't be the case if we look at this one right over here it looks like it looks like let's see if we start over here if we increase one it looked in 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 for choice be our slope is exactly 5 or 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 and so I'm going to try to find a point where it looks like I I have kind of an integer point right over here so that's the point negative 3 comma negative 3 and if I move one 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 it looks like so this one looks like it has a slope of 8 so this one does this one is increasing faster than F so I'll circle that right over there and now let's look at this choice so if we start right over here and I just pick this point because that's a nice integer coordinate it's at the point 2 comma negative 4 if we increase by 1 if we increase by 1 if we increase X by 1 then we increase Y by 1 2 3 looks like about 3 and 1/2 definitely a lot 5 in order for it to increase as fast as f it would have to increase by 5 so we'd have to be up here so we'd have 2 1 2 3 4 5 who'd have had to be up here the line would have had to look something more like would have 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