### Video Transcript

In this video, we’re gonna look at rates of change in tables of data and on
straight line graphs. In particular, we’ll see why it’s useful to come up with a way of comparing
the rates of change between two data sets, by calculating the change in one variable per unit
of another variable.

Here’s a table showing the share price of a particular company over a
four-day period. If we work out the size of the change from day to day, we can see that the
value increases by two dollars and nine cents each day. That’s
exactly the same amount each day. We call this consistent difference from day to day, a constant rate of
change.

Now if we plotted the data as a graph with the day number as the
𝑥-coordinate and the share price as the 𝑦-coordinate, we get
something like this. Now these data points are snapshots at a particular point each day, and we’ve
got no idea what happened to the prices in between times. Maybe the price went up and down in between times like this, but we don’t
have that data. Our best guess is that this looks like a nice steady constant rate of change,
which makes a straight line pattern on the graph. Without any other data to go on, this is our best guess at the behavior of
the share price in between times.

Now we can see the rate of change in the slope of the line. Every time I increase the 𝑥-coordinate by one, so
when one day passes, then the 𝑦-coordinate increases by two point zero
nine; the price goes up by two dollars and nine cents. The rate of change of the share price then would be positive two
dollars and nine cents per day. Sometimes though, we have data in a slightly more sporadic format, maybe not
every day, or week, or even year.

Barry’s height data tells us that when he was twelve years old,
he was fifty-four inches tall and when he was seventeen years old,
he was sixty-nine inches tall. Now again, we’ve got no idea how Barry’s height varied between the ages of
twelve and seventeen. Maybe he stayed fifty-four inches tall until the day before his
seventeenth birthday, and then miraculously grew fifteen inches overnight as he
turned seventeen. Yeah, it’s pretty unlikely, but really we’ve got no idea what the pattern of
growth was between the two ages. However, we could work out an average rate of growth, which in this case
would make sense to measure in inches per year. We could do some conversions and calculate the growth rate in centimetres per
month, or feet per decade, or even attometres per picosecond, if we were feeling particularly
silly, but it doesn’t really make sense. Let’s work with the data units that we’ve got.

Barry grew fifteen inches over the course of five years. An increase of fifteen inches took five years. So if we’re trying to work out a growth rate per year, we need to divide to
the number of years by five. And if we’re working out an average rate of growth, then we’re assuming a
constant rate of growth. So if they’re growing for a fifth the amount of time,
they’re gonna grow a fifth the amount of height. So in one year, they will grow fifteen divided by
five, that’s just three inches. So the rate of change of Barry’s height is positive three inches
per year. So growing at that constant rate of three inches per year, the
graph of his height would look like this. And the average rate of change of height per year is gonna tell us about the
slope of the line on our graph. When we increase the age by one year, the height goes up by
three inches, so the slope will be three, positive
three.

But rates of change become particularly useful when we’re comparing things. For example, here’s some data for two cars. It gives us the amount of fuel
that’s left in their tanks after travelling a certain number of miles. So car A,
when it traveled a hundred and fifty miles, had nine gallons of fuel
remaining. And when it traveled five hundred miles, it only had one point
two gallons of fuel remaining. Car B, when it hadn’t traveled any
distance at all, it had a tank with eleven gallons in it. And when it’d traveled
two hundred miles, there was only six point five gallons of fuel
left in the tank. Now it’s not obvious straightaway from the two tables which is the most
fuel-efficient car, the one that has the smallest rate of change of fuel left in the tank. And
that’s because they both use different amounts of fuel and travel different distances. But if we work out the average rate of change for each vehicle, then we can
make direct comparisons between them.

For car A, the distance increased a hundred and
fifty to five hundred, so that’s an increase of three hundred and
fifty miles. And in doing that three hundred and fifty miles, the amount
of fuel remaining dropped from nine to one point two gallons, which
is a drop of seven point eight gallons. In car B, traveled two hundred miles and used
three point five gallons of fuel doing it. Now we’re gonna tackle this question two different ways. First, we’re gonna look at the rate of change of fuel remaining in the tank
per mile traveled. And then we’re gonna look at the rate of change of distance traveled per
gallon of fuel used.

So for car A, when we increased the number of miles that we
traveled by three hundred and fifty, the amount of fuel remaining in the tank
dropped by seven point eight gallons. So if I want to work out by how much the fuel remaining dropped for
one mile, I would need to travel a three hundred and fiftieth of
the distance, so I need to divide both those numbers by three hundred and fifty. So for car A, when I increase the number of miles that I’ve
traveled by one, the number of gallons of fuel left in my tank drops by
nought point nought two two three gallons, to four decimal places. And for car B, in travelling two hundred miles, our
fuel drops by three point five gallons. So to work out the fuel drop for one mile, I need to divide both
by two hundred. If I only travel one two hundredth the distance, I’m only going
to use one two hundredth the amount of fuel. In this case, I get an accurate answer for the number of gallons. So when I
travel one mile in car B, the amount of fuel left in my tank drops
by nought point nought one seven five gallons. So I can see that the fuel level is changing at a rate of negative
nought point nought two two three gallons per mile in car A and
negative nought point nought one seven five gallons per mile in car
B. And this means that car B has got the smaller magnitude of
average rate of change in fuel remaining. And that means that its rate of change is closer to zero, which
means that it’s using less fuel to travel each mile on average.

If we plot both of these sets of data on the same axes, we get a good visual
representation of the difference in efficiency of the two cars. The slope of each of the lines is different, and it’s reasonably easy to say
that the amount of fuel remaining is dropping faster in car A than it is in car
B.

Okay. So now let’s go back and do the calculation again, this time working
out how many miles the car travels for each gallon of fuel that it uses. So car A traveled three hundred and fifty miles
using seven point eight gallons. Now if we divide that by seven point eight, we’re using
one seven point eighth as much fuel, so we’ll only get one seven point
eighth as far. Now we see that car A travels forty-four point eight seven
one miles with one gallon of fuel. And if we do a similar calculation for car B, we can see that
it’ll travel fifty-seven point one four three miles on one gallon of
fuel. And this way round, when we compare the fuel efficiency of the cars, we need
to find the one with the greatest rate of change, to find the most efficient. The car
travelling the furthest distance on the same one gallon of fuel is the one that’s
most efficient, and that’s car B.

So think carefully before interpreting your numbers. There are usually two
ways of approaching problems like this. And in one case, we would conclude that the greater
rate of change come with the more efficient cars. And the other way around, it’s the smaller
rate of change that tells you it’s a more efficient car. Now let’s just take a quick look at a couple of typical rates of change
questions.

Determine which of these two functions has a greater rate of change. And for
A, we’ve got a table of values. When 𝑥 is twelve,
𝑦 is three and when 𝑥 is fifty-three,
𝑦 is a hundred. And for B, we’ve been given a
straight line graph and we’ve got two points. The first point on the left, when 𝑥 is
zero, 𝑦 is one. And when 𝑥 is ten, 𝑦 is three,
in the point on the right.

With the first function A, when the 𝑥-coordinate
increases by forty-one, the 𝑦-coordinate increases by
ninety-seven. Now we can define our rate of change as, if I increase the
𝑥-coordinate by one, then how much does the
𝑦-coordinate change by? Well I can see that If I increase 𝑥 by forty-one,
then 𝑦 increases by ninety-seven. But if I only increase 𝑥 by one forty-oneth of the
amount, if I divide that by forty-one, then the corresponding
𝑦-value is only going to increase by one forty-oneth of the amount
as well. So I need to divide that ninety-seven by forty-one. So the rate of change for A is positive two point three
seven.

And for B, let’s consider the rate of change going from the
first point to the second point. If I increase my 𝑥-coordinate by
ten, then the 𝑦-coordinate increases by two. Remember, my rate of changes is, if I increase my 𝑥-coordinate
by one, what does my 𝑦-coordinate increase by? Well I’ve got the value for increasing 𝑥 by ten. I
only want to increase it by a tenth as much. And correspondingly, 𝑦 will only increase by tenth
as much as well, so two divided by ten. So the rate of change for B is positive nought point
two. And since two point three seven is greater than nought
point two, A is the function that has the greater rate of change.

Now once you understand how all of this works, there’s a bit of a shortcut
that you can take with the working out. You can calculate the rate of change in this case by doing the difference in
the 𝑦-coordinates divided by the difference in the 𝑥-coordinates. Well 𝑦 increased by ninety-seven for function
A, so a hundred minus three is ninety-seven. And the 𝑥-coordinate changed by forty-one, so
fifty-three minus twelve is forty-one. Now you need to be consistent about which 𝑥 you take away and
which 𝑦 you take away. They need to be the corresponding ones. So I’ve got the
right-hand point, and I’ve taken away the left-hand point. And I did the same thing for 𝑥 and 𝑦.

So now let’s do the same thing for the graph. And the difference in
𝑦-coordinates, the right-hand 𝑦-coordinate is three,
the left-hand is one, and three minus one is two. The 𝑦-coordinate has
increased by two. And for the 𝑥-coordinates, the right-hand
𝑥-coordinate is ten, the left-hand is zero, so it’s
gonna be ten minus zero, which is ten. The 𝑥-coordinate has increased by
ten. And two divided by ten is nought point two. So it’s your choice which of those two methods you’d use. They should both
come up with the same answer so long as you do them correctly.

Lastly then: Determine which of these two functions has a greater rate of
change. And we’ve been given a table of values for A and a graph for
B. And you’ll notice with the table of values that every time I increase my
𝑥-coordinate by one, the corresponding 𝑦-coordinate increases by two. So for A, we’re gonna use the rate of change definition. By how
much does 𝑦 change when I increase 𝑥 by one? And the
answer there is clearly positive two.

And for the graph of function B, I’m gonna use the definition,
it’s the difference in 𝑦-coordinates divided by the difference in
𝑥-coordinates. And again, I’m gonna do the right-hand point, this one, minus the left-hand
point, this one. So the right-hand 𝑦-coordinate take away the left-hand
𝑦-coordinate gives us
negative five take away four, which is negative nine. And the right-hand 𝑥-coordinate take away the left-hand
𝑥-coordinate gives us one take away negative two which is the same as one plus
two, which is three. And negative nine divided by three is negative three. So for function A we’ve got a rate of change of positive
two, and for function B we’ve got a rate of change of negative
three. So function B’s 𝑦-coordinates are changing more
rapidly than function A’s 𝑦-coordinates. When I increase the
𝑥-coordinate by one, function A’s
𝑦-coordinate would increase by two but function B’s
𝑦-coordinate would decrease by three.

But sorry folks, this is actually a bit of a trick question. We’ve just
identified that the magnitude of the rate of change for function B is greater
than that for function A. Negative three is further away from zero than
positive two. But if we look at the numbers representing the rates of change,
two is greater than negative three. So technically, the rate of change of function A is greater than
the rate of change of function B. I know it’s confusing, isn’t it?

Now don’t worry about this too much, but it’s just of minor technicality. So
in this case, with A, function A having a rate of change of
positive two and function B having a rate of change of
negative three, the magnitude of the rate of change of function B
is greater than the magnitude of the rate of change of function A because
negative three is further away from zero than positive
two. But the actual rate of change of function A is greater than the
rate of change of function B because two is greater than
negative three.

Now you’re actually quite unlikely to encounter questions like those ones
we’ve just looked at. I just wanted you to be aware of that minor technicality. So let’s move
on and do a summary of the rates of change. We can define the rate of change in a function in two ways. Firstly, by how
much does the 𝑦-coordinate change, if I increase my 𝑥-coordinate by
one. And it amounts to the same thing, but another definition is, the rate of
change is the difference in 𝑦-coordinates divided by the difference in
𝑥-coordinates. But if you use that method, you must be consistent. If you do the right-hand
𝑦-coordinate take away the left-hand 𝑦-coordinate, then you must
do the right-hand 𝑥-coordinate take away the left-hand 𝑥-coordinate.