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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. So they give us different values
of x and what the function is for each of those x's. Which graphs show
functions which are increasing at
the same rate as f? So what is the rate at
which f is increasing? When x increases by 4, we have
our function increasing by 7. So we could just look
for which of these lines are increasing at a rate of
7/4, 7 in the vertical direction every time we move 4 in
the horizontal direction. And an easy way to eyeball
that would actually be just to plot two points for f, and
then see what that rate looks like visually. So if we see here when
x is 0, f is negative 1. When x is 0, f is negative 1. So when x is 0, f is negative 1. And when x is 4, f is
6, so 1, 2, 3, 4, 5, 6, so just like that. And two points specify a line. We know that it is
a linear function. You can even verify it here. When we increase by 4 again,
we increase our function by 7 again. We know that these
two points are on f and so we get a sense
of the rate of change of f. Now, when you draw it
like that, it immediately becomes pretty
clear which of these has the same rate
of change of f. A is increasing faster than f. C is increasing slower. A is increasing
much faster than f. C is increasing slower than f. B is decreasing, so
that's not even close. But D seems to have the
exact same inclination, the exact same slope, as f. So D is what we would go with. And we could even
verify it, even if we didn't draw
it in this way. Our change in f for
a given change in x is equal to-- when
x changed plus 4, our function changed plus 7. It is equal to 7/4. And we can verify that on D, if
we increase in the x-direction by 4, so we go from 4 to 8,
then in the vertical direction we should increase by 7,
so 1, 2, 3, 4, 5, 6, 7. And it, indeed, does increase
at the exact same rate.