WEBVTT
00:00:02.240 --> 00:00:08.560
In this video, we’re going to look at how we can interpret linear functions in real-world situations.
00:00:08.820 --> 00:00:12.350
Linear functions can be seen in many different contexts.
00:00:12.600 --> 00:00:20.200
For example, in the sciences, we might see them in distance–time graphs or in converting temperatures from Celsius to Fahrenheit.
00:00:20.380 --> 00:00:27.930
They’re also seen in different business contexts, for example, the miles that a sales person might cover and the cost to the company.
00:00:28.340 --> 00:00:32.120
So let’s begin by thinking about what a linear function is.
00:00:32.670 --> 00:00:39.220
When we began studying linear functions, we saw these in the form 𝑦 equals 𝑚𝑥 plus 𝑏.
00:00:39.530 --> 00:00:42.380
There are two variables 𝑦 and 𝑥.
00:00:42.620 --> 00:00:48.170
And importantly, there’s no higher power of 𝑥 other than 𝑥 to the power of one.
00:00:48.340 --> 00:00:54.250
If we had 𝑥 squared, for example, then this would be a quadratic function and not a linear function.
00:00:54.600 --> 00:01:02.480
The value of 𝑚 represents the slope or gradient of the function, and 𝑏 represents the 𝑦-intercept of the function.
00:01:02.730 --> 00:01:08.740
Sometimes this can be interchanged with the letter 𝑐 to give us the linear function 𝑦 equals 𝑚𝑥 plus 𝑐.
00:01:09.030 --> 00:01:12.720
But either way, the 𝑏 or the 𝑐 represents the 𝑦-intercept.
00:01:13.370 --> 00:01:21.850
If we were to graph this linear function 𝑦 equals 𝑚𝑥 plus 𝑏, then the 𝑏 would represent the point where the line crosses the 𝑦-axis.
00:01:22.250 --> 00:01:32.580
But of course, when it comes to linear functions in a real-world context, the equations or functions that we’re given would always be in this nice, easy-to-use format.
00:01:32.970 --> 00:01:39.530
But we know that if the problem involves a constant rate of change, then it will be a linear function.
00:01:39.770 --> 00:01:45.300
We can use what we know about slope and the 𝑦-intercept to fully investigate the problem.
00:01:45.610 --> 00:01:50.850
In the first few questions that we look at, we’re not going to worry about the graph of the function.
00:01:51.070 --> 00:01:57.030
Instead, we’re going to take a real-world problem and try to write it in this linear function form.
00:01:57.410 --> 00:02:04.580
In our questions, the variables won’t be given as 𝑥 and 𝑦 but will be related to the context of the problem.
00:02:04.880 --> 00:02:07.290
So let’s have a look at our first question.
00:02:09.800 --> 00:02:14.610
In 1995, music stores sold cassette tapes for two dollars.
00:02:14.910 --> 00:02:24.460
Write an equation to find 𝑡, the total cost in dollars for buying 𝑐 cassette tapes, and then find out how much it would cost to buy three cassette tapes.
00:02:25.090 --> 00:02:29.620
In this question, we’re given the information about the cost of a cassette tape.
00:02:29.890 --> 00:02:33.390
The next part of this question might seem quite confusing.
00:02:33.680 --> 00:02:40.640
We’re told to write an equation to find 𝑡, the total cost in dollars, for buying 𝑐 cassette tapes.
00:02:41.030 --> 00:02:42.600
So let’s break this down.
00:02:42.800 --> 00:02:55.640
We’re told that one cassette tape would cost two dollars, which means that two cassette tapes would cost four dollars and three cassette tapes would cost six dollars and so on.
00:02:56.010 --> 00:03:01.470
So how about if we bought 𝑐 cassette tapes where we don’t know the value of 𝑐?
00:03:01.700 --> 00:03:03.530
How much would that cost?
00:03:04.020 --> 00:03:08.250
Well, each cassette tape is still going to cost two dollars.
00:03:08.530 --> 00:03:14.590
So the total cost would be two times 𝑐 dollars or simply two 𝑐 dollars.
00:03:15.030 --> 00:03:19.160
We could then say that the total cost would be two 𝑐 dollars.
00:03:19.410 --> 00:03:23.800
But, however, we were asked to find 𝑡, the cost in dollars.
00:03:24.050 --> 00:03:28.750
This means that we need to replace the wording of total cost with 𝑡.
00:03:29.200 --> 00:03:35.290
We can also get rid of the dollar sign as we’re told that 𝑡 is the cost in dollars.
00:03:35.600 --> 00:03:39.770
We have now written an equation in the variables of 𝑡 and 𝑐.
00:03:40.200 --> 00:03:46.880
Notice that the two in this equation represents the cost of each cassette and its constant.
00:03:47.110 --> 00:03:55.310
If we compare this to the linear function 𝑦 equals 𝑚𝑥 plus 𝑏, then we might notice that we don’t have any 𝑏-value.
00:03:55.640 --> 00:04:02.230
If we were to graph this function of 𝑐 against 𝑡, then the graph would look like this.
00:04:02.560 --> 00:04:08.240
The slope of the line would be two as one cassette tape costs two dollars.
00:04:08.530 --> 00:04:21.070
And the 𝑦-intercept would be zero because if we bought zero cassette tapes, it would cost us zero dollars which is why the 𝑏-value in the linear function form is zero.
00:04:21.630 --> 00:04:28.200
Now that we found our equation, we’re asked for one more thing, the cost of buying three cassette tapes.
00:04:28.530 --> 00:04:33.080
We can use our equation plugging in the value of 𝑐 equals three.
00:04:33.430 --> 00:04:42.880
This would give us 𝑡 equals two times three, which of course would give us six, which means that three cassette tapes would cost six dollars.
00:04:43.270 --> 00:04:45.360
We can then give our two answers.
00:04:45.600 --> 00:04:52.160
The equation we found is 𝑡 equals two 𝑐 and the cost of three cassette tapes is six dollars.
00:04:53.920 --> 00:04:55.990
Let’s look at another question.
00:04:58.140 --> 00:05:00.990
Sophia has 10 dollars in her bank account.
00:05:01.320 --> 00:05:04.510
Every week she will deposit 20 dollars into the account.
00:05:04.790 --> 00:05:11.410
Write an equation that represents this situation, where 𝑇 is the total money in her account after 𝑤 weeks.
00:05:11.980 --> 00:05:16.420
Let’s begin this question by thinking about the money in Sophia’s bank account.
00:05:16.850 --> 00:05:21.380
At the start of this problem, Sophia has 10 dollars in her bank account.
00:05:21.640 --> 00:05:25.040
We’re told that she adds 20 dollars every week.
00:05:25.240 --> 00:05:30.150
So in the first week, she’ll have the 10 dollars plus another 20 dollars.
00:05:30.380 --> 00:05:35.100
In week two, she’ll have the 10 dollars plus two lots of 20 dollars.
00:05:35.350 --> 00:05:39.910
In week three, she’ll have 10 dollars and three lots of 20 dollars.
00:05:40.120 --> 00:05:46.080
We could also think of this week three as 10 plus three times 20 dollars.
00:05:46.230 --> 00:05:50.510
We could continue this pattern adding 20 dollars every week.
00:05:50.810 --> 00:05:56.270
This is where the variables of 𝑇 and 𝑤 will actually be quite helpful.
00:05:56.500 --> 00:06:00.370
Somewhere along in this pattern, we will have week 𝑤.
00:06:00.710 --> 00:06:03.530
We would still have 10 dollars in the account.
00:06:03.760 --> 00:06:11.250
And although we don’t know quite how many 20 dollars we’ll have, we know that it would be equal to 𝑤 times 20.
00:06:11.460 --> 00:06:16.250
So the total in the account would be 10 plus 𝑤 times 20.
00:06:16.580 --> 00:06:21.690
We can’t just give this as our answer, however, as we’re asked to write an equation.
00:06:21.910 --> 00:06:24.760
And we’ll do this also using the letter 𝑇.
00:06:25.170 --> 00:06:34.590
As 𝑇 is the total money in the account after 𝑤 weeks, we could write this as 𝑤 times 20 or 20𝑤 plus 10.
00:06:34.850 --> 00:06:37.340
And this will be our answer for the equation.
00:06:37.480 --> 00:06:44.360
It would still have been mathematically correct to give this answer as 𝑇 equals 10 plus 20𝑤.
00:06:44.790 --> 00:06:49.660
Before we finish this question, let’s have a closer look at the equation that we’ve written.
00:06:49.930 --> 00:06:55.080
If we were to draw a graph of this linear function, it would be a straight line graph.
00:06:55.240 --> 00:07:03.660
The value of 20 represents a constant change, and that’s because every week there were 20 dollars added to Sophia’s account.
00:07:03.820 --> 00:07:14.380
The value of 10 would be the 𝑦-intercept, this value on the graph, which show that, at week zero or the start, there were 10 dollars in the bank account.
00:07:14.690 --> 00:07:18.310
And so we can see how we’ve created a linear equation.
00:07:20.010 --> 00:07:25.060
In the next question, we’ll see how we can interpret the graph of a linear function.
00:07:27.770 --> 00:07:32.050
An electrician charges a call-out fee and an hourly labor charge.
00:07:32.350 --> 00:07:38.140
The graph represents what the electrician charges in dollars for jobs of different durations.
00:07:38.400 --> 00:07:40.970
What is the electrician’s call-out fee?
00:07:41.220 --> 00:07:43.470
What is the hourly labor charge?
00:07:43.760 --> 00:07:48.110
Let 𝑦 be the cost in dollars for a job that takes 𝑥 hours.
00:07:48.400 --> 00:07:51.530
Write an equation for 𝑦 in terms of 𝑥.
00:07:52.390 --> 00:08:06.240
Let’s begin by having a look at the graph of this function, we can see along the 𝑥-axis that we have the time in hours that the electrician spends on the job and the 𝑦-axis represents the cost in dollars.
00:08:06.650 --> 00:08:11.610
The graph is a straight line, so we know that this would be a linear function.
00:08:11.840 --> 00:08:15.930
In a linear graph, there will be a constant rate of change.
00:08:16.210 --> 00:08:22.180
What we need to do here is extract the information about the call-out fee and the hourly labor charge.
00:08:22.480 --> 00:08:24.980
Let’s begin with the call-out fee.
00:08:25.300 --> 00:08:27.820
So what exactly is a call-out fee?
00:08:28.210 --> 00:08:35.670
The call-out fee is the part of the bill that the electrician would make regardless of how long they spend fixing the problem.
00:08:35.930 --> 00:08:44.270
For example, if they turned up to see a problem and they couldn’t fix it, they’d still charge the call-out fee just for turning up.
00:08:44.460 --> 00:08:50.150
The call-out fee on this graph would be represented at the point when the time is equal to zero.
00:08:50.500 --> 00:08:56.440
That’s the basic amount the electrician charges even if they don’t spend any time fixing the problem.
00:08:56.670 --> 00:08:59.090
We can then read off 40 on the graph.
00:08:59.370 --> 00:09:04.060
And as we know that the cost is given in dollars, this will be 40 dollars.
00:09:04.520 --> 00:09:07.610
And that’s the answer for the first part of the question.
00:09:07.930 --> 00:09:14.660
In the second question, we’re asked for the hourly labor charge or the amount the electrician charges each hour.
00:09:14.910 --> 00:09:20.930
If we look on the graph, at one hour, we can see that the charge here would be 100 dollars.
00:09:21.280 --> 00:09:23.750
So is that the answer to the second question?
00:09:24.070 --> 00:09:25.480
Well, not quite.
00:09:25.910 --> 00:09:32.180
If it was, we’d expect that after another hour, the charge would be another 100 dollars.
00:09:32.560 --> 00:09:36.540
Looking at the graph, we can see that this wouldn’t be the case.
00:09:36.790 --> 00:09:41.640
We can in fact find the hourly labor charge by finding the slope of the line.
00:09:41.900 --> 00:09:47.230
And we do this by using the formula that the slope is equal to the rise over the run.
00:09:47.550 --> 00:09:54.520
If we select any two points on the line, we can find the rise by finding how much the graph has gone up.
00:09:54.800 --> 00:09:58.370
Between our two points, it will have gone up by 60.
00:09:58.800 --> 00:10:02.090
The run will be the change in our 𝑥-values.
00:10:02.090 --> 00:10:05.950
Here, that would be two subtract one, which is one.
00:10:06.330 --> 00:10:11.570
And so the slope is equal to 60 over one, which is equal to 60.
00:10:11.880 --> 00:10:15.070
The slope of this line will always be 60.
00:10:15.260 --> 00:10:23.150
In the context of the problem, that means the electrician is charging 60 dollars for every hour they spend working on the problem.
00:10:23.340 --> 00:10:29.770
We can therefore give our answer to the second part of this question that the hourly labor charge is 60 dollars.
00:10:30.230 --> 00:10:35.270
To answer the third part of this question, we’re going to see two alternative methods.
00:10:35.330 --> 00:10:40.920
The first method involves looking at the problem and then trying to write a mathematical statement.
00:10:41.140 --> 00:10:46.370
We can start by thinking what the total cost would be for an electrician’s time.
00:10:46.630 --> 00:10:51.250
We know that the electrician will start by always charging the call-out fee.
00:10:51.400 --> 00:10:58.190
Then they add on the hourly rate multiplied by the number of hours that the electrician is working.
00:10:58.370 --> 00:11:02.860
We can rewrite this in a nicer way using the variables that we’re given.
00:11:03.220 --> 00:11:09.780
We’re told that 𝑦 should be the cost in dollars, so we can begin our statement with 𝑦 equals.
00:11:10.150 --> 00:11:13.360
We know that the call-out fee is 40 dollars.
00:11:13.780 --> 00:11:17.970
And we add on the hourly rate times the number of hours.
00:11:18.170 --> 00:11:21.280
We know that the hourly labor charge is 60 dollars.
00:11:21.480 --> 00:11:24.250
And we’re told to use 𝑥 for the number of hours.
00:11:24.540 --> 00:11:31.960
We could then give our answer either as 𝑦 equals 40 plus 60𝑥 or as 𝑦 equals 60𝑥 plus 40.
00:11:32.480 --> 00:11:44.420
As an alternative method, we could take the approach that, as we know, that this is a linear function, then it must conform to 𝑦 equals 𝑚𝑥 plus 𝑏, which is the general form of a linear function.
00:11:44.630 --> 00:11:50.940
We can remember that the 𝑚-value represents the slope and the 𝑏-value represents the 𝑦-intercept.
00:11:51.500 --> 00:11:59.580
In the second part of our question, we worked out that the hourly labor charge is 60 dollars and that was the slope of the line.
00:11:59.810 --> 00:12:07.380
So our equation would begin 𝑦 equals 60𝑥 plus the 𝑦-intercept, which would be 40.
00:12:07.970 --> 00:12:13.520
Either method would give the answer for the equation 𝑦 equals 60𝑥 plus 40.
00:12:15.380 --> 00:12:17.850
Let’s look at one final question.
00:12:20.170 --> 00:12:37.880
Suppose that the average annual income in dollars for the years 1990 through 1999 is given by the linear function 𝐼 of 𝑥 equals 1,054𝑥 plus 23,286, where 𝑥 is the number of years after 1990.
00:12:38.210 --> 00:12:42.470
Which of the following interprets the slope in the context of the problem?
00:12:42.860 --> 00:12:51.300
Option (A) the average annual income rose to level of 23,286 by the end of 1999.
00:12:51.570 --> 00:13:00.120
Option (B) each year in the decade of the 1990s, the average annual income increased by 1,054 dollars.
00:13:00.540 --> 00:13:10.600
Option (C) in the 10-year period from 1990 to 1999, the average annual income increased by a total of 1,054 dollars.
00:13:10.960 --> 00:13:19.160
Option (D) as of 1990, the average annual income was 23,286 dollars.
00:13:20.040 --> 00:13:33.370
As we start a question like this, it’s important to realize that even though of course we need a good level of mathematical knowledge, there’s also a level of linguistic understanding that we need to understand the language.
00:13:33.990 --> 00:13:37.250
Let’s pull out the key information that we’re given.
00:13:37.510 --> 00:13:47.550
We’re told that the average annual income is given by this function 1,054𝑥 plus 23,286.
00:13:47.960 --> 00:13:56.090
We’re told that this is a linear function, which means that it fits into the form of 𝑦 equals 𝑚𝑥 plus 𝑏.
00:13:56.460 --> 00:14:05.870
When we’re looking at linear functions, the value of 𝑚 represents the slope or gradient and the value of 𝑏 represents the 𝑦-intercept.
00:14:06.540 --> 00:14:09.740
In the question, we’re asked to interpret the slope.
00:14:09.990 --> 00:14:18.830
So what we’re really looking at is what does this value of 1,054 actually mean in terms of the average annual income?
00:14:19.210 --> 00:14:21.830
Now, we know that this is a linear function.
00:14:22.060 --> 00:14:25.070
So let’s see if we could model this graph.
00:14:25.330 --> 00:14:31.160
We can plot the years along the 𝑥-axis and the income in dollars on the 𝑦-axis.
00:14:31.580 --> 00:14:37.520
The years that we’re told is between the time period of 1990 and 1999.
00:14:37.850 --> 00:14:41.210
We really don’t have to worry about being superaccurate.
00:14:41.420 --> 00:14:43.970
This is just going to be a sketch of the graph.
00:14:44.300 --> 00:14:47.460
But let’s begin by thinking about the 𝑦-intercept.
00:14:47.900 --> 00:14:56.290
The 𝑏-value in our linear function will be the 𝑦-intercept, which is this value of 23,286.
00:14:56.540 --> 00:14:59.250
So this is where our graph will cross the 𝑦-axis.
00:14:59.570 --> 00:15:06.520
The gradient 𝑚 is a positive value, so we know that our line is going to slope upwards from left to right?
00:15:06.990 --> 00:15:10.200
Let’s think a little bit more about the slope of the line.
00:15:10.530 --> 00:15:14.180
The slope is given here as 1,054.
00:15:14.450 --> 00:15:21.010
That means that, for every year, the income will increase by 1,054 dollars.
00:15:21.350 --> 00:15:26.530
Because this is a linear function, that means that there’s a constant rate of change.
00:15:26.770 --> 00:15:32.890
In other words, every year, this income will increase by 1,054 dollars.
00:15:33.380 --> 00:15:46.400
If we have a look at our answer statements, then the one that fits would be answer (B), each year in the decade of the 1990s, the average annual income increased by 1,054 dollars.
00:15:47.000 --> 00:15:52.980
So let’s take a look at some of the other answer options to see why these wouldn’t be correct.
00:15:53.710 --> 00:16:02.020
Option (A) says the average annual income rose to level of 23,286 by the end of 1999.
00:16:02.320 --> 00:16:04.900
So let’s see what that would look like on a graph.
00:16:05.130 --> 00:16:13.160
It would mean that in 1999, we would have an average annual income rating at 23,286.
00:16:13.380 --> 00:16:20.520
And we don’t have this as we know that the graph begins at 23,286 in 1990.
00:16:21.010 --> 00:16:23.090
So option (A) is incorrect.
00:16:23.590 --> 00:16:33.830
Option (C) says in the 10-year period from 1990 to 1999, the average annual income increased by a total of 1,054 dollars.
00:16:34.220 --> 00:16:40.020
It would be very easy to think that this really does sound very similar to the actual answer in option (B).
00:16:40.260 --> 00:16:42.350
But there are important differences.
00:16:42.640 --> 00:16:50.730
In this case, they’re saying that in the whole 10 years then the average increased by a total of 1,054.
00:16:51.050 --> 00:16:57.260
This would mean that the increase would be 1,054 over the whole 10 years.
00:16:57.440 --> 00:17:00.250
And that is not what we found on our graph.
00:17:00.680 --> 00:17:09.220
Finally, let’s look at option (D), as of 1990, the average annual income was 23,286.
00:17:09.610 --> 00:17:19.640
This statement would imply that even though we do have 1990 is this value, it would seem that this value doesn’t change across the 10-year period.
00:17:19.850 --> 00:17:26.870
As this is not the case, then we can definitely eliminate option (D), leaving us with our answer of option (B).
00:17:28.610 --> 00:17:31.620
We can now summarize what we’ve learned in this video.
00:17:32.320 --> 00:17:36.300
We recalled, firstly, that a linear function is a straight line.
00:17:36.450 --> 00:17:42.770
Its general form is 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏 is the 𝑦-intercept.
00:17:43.430 --> 00:17:47.780
We saw that linear functions have a constant rate of change.
00:17:48.260 --> 00:17:55.940
As we saw in a number of questions, often, the variables 𝑥 and 𝑦 will be given different labels depending on the context.