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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.