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CCSS Math: 8.F.A.2, HSF.IF.C.9

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.