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
2.09. The price goes up by two dollars
and nine cents. The rate of change of the share
price then will 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 54 inches tall and when he was 17 years old, he
was 69 inches tall.
Now again, we’ve got no idea how
Barry’s height varied between the ages of 12 and 17. Maybe, he stayed 54 inches tall
until the day before his 17th birthday and then miraculously grew 15 inches
overnight as he turned 17. 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
Barry grew 15 inches over the
course of five years. An increase of 15 inches took five
years. So if we’re trying to work out a
growth rate per year, we need to divide 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 15
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,
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 150
miles, had nine gallons of fuel remaining. And when it traveled 500 miles, it
only had 1.2 gallons of fuel remaining. Car B, when it hadn’t traveled any
distance at all, it had a tank with 11 gallons in it. And when it’d traveled 200 miles,
there was only 6.5 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 used
different amounts of fuel and traveled 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
150 to 500, so that’s an increase of 350 miles. And in doing that 350 miles, the
amount of fuel remaining dropped from nine to 1.2 gallons, which is a drop of 7.8
gallons. In car B, traveled 200 miles and
used 3.5 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 350, the amount of fuel remaining in the tank
dropped by 7.8 gallons. So if I want to work out by how
much the fuel remaining dropped for one mile, I would need to travel a 350th of the
distance. So I need to divide both those
numbers by 350. 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 0.0223 gallons to four decimal places. And for car B, in travelling 200
miles, our fuel drops by 3.5 gallons. So to work out the fuel drop for
one mile, I need to divide both by 200. If I only travel one 200th the
distance, I’m only going to use one 200th 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 0.0175 gallons.
So I can say that the fuel level is
changing at a rate of negative 0.0223 gallons per mile in car A and negative 0.0175
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
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 see
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 350 miles using
7.8 gallons. Now we divide that by 7.8. We’re using one 7.8th as much
fuel. So we’ll only get one 7.8th as
far. Now we see that car A travels
44.871 miles with one gallon of fuel. And if we do a similar calculation
for car B, we can see that it’ll travel 57.143 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 12, 𝑦 is three. And when 𝑥 is 53, 𝑦 is 100. 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 10, 𝑦 is three, in
the point on the right.
With the first function A, when the
𝑥-coordinate increases by 41, the 𝑦-coordinate increases by 97. 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 41, then 𝑦 increases by 97. But if I only increase 𝑥 by one
41th of the amount, if I divide that by 41, then the corresponding 𝑦-value is only
going to increase by one 41th of the amount as well. So I need to divide that 97 by
41. So the rate of change for A is
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
10, then the 𝑦-coordinate increases by two. Remember, my rate of change 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 10. 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 10. So the rate of change for B is
positive 0.2. And since 2.37 is greater than 0.2,
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 97 for
function A. So 100 minus three is 97. And the 𝑥-coordinate changed by
41. So 53 minus 12 is 41.
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 10. The left-hand is zero. So it’s gonna be 10 minus zero,
which is 10. The 𝑥-coordinate has increased by
10. And two divided by 10 is 0.2. So it’s your choice which of those
two methods you’ll 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.