Lesson Video: Applications of Linear Functions Mathematics

Video Transcript

In this video, we’re going to look at how we can interpret linear functions in real-world situations. Linear functions can be seen in many different contexts. For example, in the sciences, we might see them in distance–time graphs or in converting temperatures from Celsius to Fahrenheit. They’re also seen in different business contexts, for example, the miles that a sales person might cover and the cost to the company.

So let’s begin by thinking about what a linear function is. When we began studying linear functions, we saw these in the form 𝑦 equals 𝑚𝑥 plus 𝑏. There are two variables 𝑦 and 𝑥. And importantly, there’s no higher power of 𝑥 other than 𝑥 to the power of one. If we had 𝑥 squared, for example, then this would be a quadratic function and not a linear function. The value of 𝑚 represents the slope or gradient of the function, and 𝑏 represents the 𝑦-intercept of the function. Sometimes this can be interchanged with the letter 𝑐 to give us the linear function 𝑦 equals 𝑚𝑥 plus 𝑐. But either way, the 𝑏 or the 𝑐 represents the 𝑦-intercept.

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. 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. But we know that if the problem involves a constant rate of change, then it will be a linear function. We can use what we know about slope and the 𝑦-intercept to fully investigate the problem.

In the first few questions that we look at, we’re not going to worry about the graph of the function. Instead, we’re going to take a real-world problem and try to write it in this linear function form. In our questions, the variables won’t be given as 𝑥 and 𝑦 but will be related to the context of the problem. So let’s have a look at our first question.

In 1995, music stores sold cassette tapes for two dollars. 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.

In this question, we’re given the information about the cost of a cassette tape. The next part of this question might seem quite confusing. We’re told to write an equation to find 𝑡, the total cost in dollars, for buying 𝑐 cassette tapes. So let’s break this down. 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.

So how about if we bought 𝑐 cassette tapes where we don’t know the value of 𝑐? How much would that cost? Well, each cassette tape is still going to cost two dollars. So the total cost would be two times 𝑐 dollars or simply two 𝑐 dollars. We could then say that the total cost would be two 𝑐 dollars. But, however, we were asked to find 𝑡, the cost in dollars. This means that we need to replace the wording of total cost with 𝑡. We can also get rid of the dollar sign as we’re told that 𝑡 is the cost in dollars. We have now written an equation in the variables of 𝑡 and 𝑐.

Notice that the two in this equation represents the cost of each cassette and its constant. 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. If we were to graph this function of 𝑐 against 𝑡, then the graph would look like this. The slope of the line would be two as one cassette tape costs two dollars. 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. Now that we found our equation, we’re asked for one more thing, the cost of buying three cassette tapes.

We can use our equation plugging in the value of 𝑐 equals three. 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. We can then give our two answers. The equation we found is 𝑡 equals two 𝑐 and the cost of three cassette tapes is six dollars.

Let’s look at another question.

Sophia has 10 dollars in her bank account. Every week she will deposit 20 dollars into the account. Write an equation that represents this situation, where 𝑇 is the total money in her account after 𝑤 weeks.

Let’s begin this question by thinking about the money in Sophia’s bank account. At the start of this problem, Sophia has 10 dollars in her bank account. We’re told that she adds 20 dollars every week. So in the first week, she’ll have the 10 dollars plus another 20 dollars. In week two, she’ll have the 10 dollars plus two lots of 20 dollars. In week three, she’ll have 10 dollars and three lots of 20 dollars. We could also think of this week three as 10 plus three times 20 dollars. We could continue this pattern adding 20 dollars every week. This is where the variables of 𝑇 and 𝑤 will actually be quite helpful.

Somewhere along in this pattern, we will have week 𝑤. We would still have 10 dollars in the account. 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. So the total in the account would be 10 plus 𝑤 times 20. We can’t just give this as our answer, however, as we’re asked to write an equation. And we’ll do this also using the letter 𝑇. As 𝑇 is the total money in the account after 𝑤 weeks, we could write this as 𝑤 times 20 or 20𝑤 plus 10. And this will be our answer for the equation. It would still have been mathematically correct to give this answer as 𝑇 equals 10 plus 20𝑤.

Before we finish this question, let’s have a closer look at the equation that we’ve written. If we were to draw a graph of this linear function, it would be a straight line graph. The value of 20 represents a constant change, and that’s because every week there were 20 dollars added to Sophia’s account. 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. And so we can see how we’ve created a linear equation.

In the next question, we’ll see how we can interpret the graph of a linear function.

An electrician charges a call-out fee and an hourly labor charge. The graph represents what the electrician charges in dollars for jobs of different durations. What is the electrician’s call-out fee? What is the hourly labor charge? Let 𝑦 be the cost in dollars for a job that takes 𝑥 hours. Write an equation for 𝑦 in terms of 𝑥.

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. The graph is a straight line, so we know that this would be a linear function. In a linear graph, there will be a constant rate of change. What we need to do here is extract the information about the call-out fee and the hourly labor charge.

Let’s begin with the call-out fee. So what exactly is a call-out fee? The call-out fee is the part of the bill that the electrician would make regardless of how long they spend fixing the problem. 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. The call-out fee on this graph would be represented at the point when the time is equal to zero. That’s the basic amount the electrician charges even if they don’t spend any time fixing the problem. We can then read off 40 on the graph. And as we know that the cost is given in dollars, this will be 40 dollars. And that’s the answer for the first part of the question.

In the second question, we’re asked for the hourly labor charge or the amount the electrician charges each hour. If we look on the graph, at one hour, we can see that the charge here would be 100 dollars. So is that the answer to the second question? Well, not quite. If it was, we’d expect that after another hour, the charge would be another 100 dollars. Looking at the graph, we can see that this wouldn’t be the case. We can in fact find the hourly labor charge by finding the slope of the line. And we do this by using the formula that the slope is equal to the rise over the run.

If we select any two points on the line, we can find the rise by finding how much the graph has gone up. Between our two points, it will have gone up by 60. The run will be the change in our 𝑥-values. Here, that would be two subtract one, which is one. And so the slope is equal to 60 over one, which is equal to 60. The slope of this line will always be 60. In the context of the problem, that means the electrician is charging 60 dollars for every hour they spend working on the problem. We can therefore give our answer to the second part of this question that the hourly labor charge is 60 dollars.

To answer the third part of this question, we’re going to see two alternative methods. The first method involves looking at the problem and then trying to write a mathematical statement. We can start by thinking what the total cost would be for an electrician’s time. We know that the electrician will start by always charging the call-out fee. Then they add on the hourly rate multiplied by the number of hours that the electrician is working. We can rewrite this in a nicer way using the variables that we’re given.

We’re told that 𝑦 should be the cost in dollars, so we can begin our statement with 𝑦 equals. We know that the call-out fee is 40 dollars. And we add on the hourly rate times the number of hours. We know that the hourly labor charge is 60 dollars. And we’re told to use 𝑥 for the number of hours. We could then give our answer either as 𝑦 equals 40 plus 60𝑥 or as 𝑦 equals 60𝑥 plus 40.

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. We can remember that the 𝑚-value represents the slope and the 𝑏-value represents the 𝑦-intercept. 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. So our equation would begin 𝑦 equals 60𝑥 plus the 𝑦-intercept, which would be 40. Either method would give the answer for the equation 𝑦 equals 60𝑥 plus 40.

Let’s look at one final question.

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. Which of the following interprets the slope in the context of the problem? Option (A) the average annual income rose to level of 23,286 by the end of 1999. Option (B) each year in the decade of the 1990s, the average annual income increased by 1,054 dollars. Option (C) in the 10-year period from 1990 to 1999, the average annual income increased by a total of 1,054 dollars. Option (D) as of 1990, the average annual income was 23,286 dollars.

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. Let’s pull out the key information that we’re given. We’re told that the average annual income is given by this function 1,054𝑥 plus 23,286. We’re told that this is a linear function, which means that it fits into the form of 𝑦 equals 𝑚𝑥 plus 𝑏. 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.

In the question, we’re asked to interpret the slope. So what we’re really looking at is what does this value of 1,054 actually mean in terms of the average annual income? Now, we know that this is a linear function. So let’s see if we could model this graph. We can plot the years along the 𝑥-axis and the income in dollars on the 𝑦-axis. The years that we’re told is between the time period of 1990 and 1999. We really don’t have to worry about being superaccurate. This is just going to be a sketch of the graph.

But let’s begin by thinking about the 𝑦-intercept. The 𝑏-value in our linear function will be the 𝑦-intercept, which is this value of 23,286. So this is where our graph will cross the 𝑦-axis. The gradient 𝑚 is a positive value, so we know that our line is going to slope upwards from left to right? Let’s think a little bit more about the slope of the line. The slope is given here as 1,054. That means that, for every year, the income will increase by 1,054 dollars. Because this is a linear function, that means that there’s a constant rate of change. In other words, every year, this income will increase by 1,054 dollars.

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.

So let’s take a look at some of the other answer options to see why these wouldn’t be correct. Option (A) says the average annual income rose to level of 23,286 by the end of 1999. So let’s see what that would look like on a graph. It would mean that in 1999, we would have an average annual income rating at 23,286. And we don’t have this as we know that the graph begins at 23,286 in 1990. So option (A) is incorrect.

Option (C) says in the 10-year period from 1990 to 1999, the average annual income increased by a total of 1,054 dollars. It would be very easy to think that this really does sound very similar to the actual answer in option (B). But there are important differences. In this case, they’re saying that in the whole 10 years then the average increased by a total of 1,054. This would mean that the increase would be 1,054 over the whole 10 years. And that is not what we found on our graph.

Finally, let’s look at option (D), as of 1990, the average annual income was 23,286. 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. As this is not the case, then we can definitely eliminate option (D), leaving us with our answer of option (B).

We can now summarize what we’ve learned in this video. We recalled, firstly, that a linear function is a straight line. Its general form is 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏 is the 𝑦-intercept. We saw that linear functions have a constant rate of change. As we saw in a number of questions, often, the variables 𝑥 and 𝑦 will be given different labels depending on the context.