### 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.