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.