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
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.