What is the rate of change shown by
this graph of a function?
Well for straight line functions,
the rate of change is defined as, by how much does the 𝑦-coordinate change when I
increase the 𝑥-coordinate by one?
Now at first glance, that’s
actually quite tricky to work out on this particular graph. So for e𝑥ample, if I go from here
where my 𝑦-coordinate is this, to increase my 𝑥-coordinate by one, so up to here
where my 𝑦-coordinate is this, this difference here is the rate of change of my
line. Now obviously, I’m not able to be
very accurate about that. But if we look closely at the
graph, they’ve helpfully put a couple of little green dots on here. So these are where the line passes
through e𝑥act coordinate pairs. So the left-hand point there has an
𝑥-coordinate of negative five and a 𝑦-coordinate of negative three. And the second point has an
𝑥-coordinate of zero and a 𝑦-coordinate of negative one.
Now in moving from the first point
to the second point, we can see that the 𝑥-coordinate here has increased by five,
and the 𝑦-coordinate here has increased by two. So let’s write that down. When 𝑥 increases by five, then 𝑦
increases by two. Now the definition of rate of
change is, by what does 𝑦 increase when 𝑥 increases by one.
So in our e𝑥ample, we’ve increased
the 𝑥-coordinate five times too much. So if we divide that by five, so
five divided by five is equal to one, so we’re going a fifth as far in the
𝑥-direction. And because it’s a straight line,
because it’s a linear relationship, because it’s proportional, this means that we’re
going o- to only go a fifth as far in the 𝑦-direction as well. So the change in 𝑦-coordinate is
the original two and we’re going to divide that by five. So rather than calculate that as a
decimal, we just literally write it as two divided by five, two fifths.
Now this rate of change is
sometimes called the slope of the line, or the gradient of the line. But in this question, we called it
rate of change. So the answer is that the rate of
change is two over five, or two fifths.