In this video, we’re gonna use our knowledge about how to represent linear functions algebraically, graphically, in tables of values, all function tables, and through verbal descriptions in order to compare different linear functions. First thought, let’s recap the basic skills.
So we’ve got a graph here which shows the population density of a particular city. 𝑥 represents the distance from the city centre, so the number of miles from the centre of the city, and 𝑦 represents the population density that’s in thousands of people per square mile.
So first let’s create an equation for the line in this graph. And there are two things that we need to know: first, we need to know what’s the 𝑦-coordinate when the 𝑥-coordinate is zero; in other words, this point here; where does it cut the 𝑦-axis?
Now remember the general format of an equation of a straight line is 𝑦 equals 𝑚𝑥 plus 𝑏. So the 𝑚, remember, is the slope of the line and the 𝑏 value here, this number that we’re adding to this multiplier times 𝑥, tells us where it cuts the 𝑦-axis. So that’s the 𝑦-intercept. So we’ve just worked that out; that is ten in this case, so 𝑏 is equal to ten.
So now we need to work out the slope of the line. And to do that, we need to work out by how much does the 𝑦-coordinate change when the 𝑥-coordinate changes by one. And if we look at the graph, when our 𝑥-coordinate goes from zero up to seven, the 𝑥-coordinate has increased by seven. And in that time, the 𝑦-coordinate has dropped from ten down to three. So that’s dropped by seven.
So when the 𝑥-coordinate increases by seven the 𝑦-coordinate drops by seven. But the definition of slope is how much does the 𝑦-coordinate change when the 𝑥-coordinate increases by only one. So if I divide the change in 𝑥-coordinate by seven, I’ve got an increase of one. So if I divide the 𝑥-coordinate by seven, I’ve only going a seventh this far across I am only going to come a seventh this far down So when I go cross one, I’m only coming down one, so the gradient is negative one.
So plugging these back into equation, it becomes 𝑦 is equal to negative one 𝑥 plus ten. And in fact negative one, we wouldn’t normally bother writing the one. So the equation becomes 𝑦 equals negative 𝑥 plus ten.
So for example, when we’re three miles from the city centre, 𝑥 would be equal to three. So plugging that into our equation, we’ve got 𝑦 is equal to negative three plus ten, which is seven. And remember, 𝑦 represents the thousands of people per square mile, so three miles away from the city centre there are seven thousand people per square mile in this city. Now let’s just go back to our graph and check that. So three miles away from the city, centre there are, if we go across here, seven thousand people per square mile, so our equation matches up with the graph.
Now secondly, we’re gonna create a table of values. And we can do this by either reading off all of these coordinates from our graph or we could plug values into our equation here and work the values out that way. So I’m gonna take a nice steadily increasing set of 𝑥-coordinates and then calculate the corresponding 𝑦-coordinates.
So I’ve just read them off of the graph. But as I say, I could’ve calculated those values by plugging the 𝑥-coordinates into my equation. So when 𝑥 is zero, 𝑦 is equal to negative zero plus ten, so that’s zero plus ten which is ten. So that’s how I got the first one; and when 𝑥 is one, 𝑦 is equal to negative one plus ten, so that’s equal to nine. And that’s how I got second one, and so on.
And we can consider rates of change from our table of values just like we did with the equation. So I’ve got steadily increasing 𝑥-coordinates; in fact, they’re all going up by one. So if I look at the change in the 𝑦-coordinate that corresponds to that, every time the 𝑥-coordinate increases by one, so going from zero to one for example, the corresponding 𝑦-coordinate is decreasing by one, for example going down from ten to nine. And that’s the case in all of these different pairs of coordinates.
And now thirdly, we’re gonna put all of this into words. We’re gonna describe the rate of change of the population density as the distance from the city centre varies. Now the fact that this graph is a straight line graph shows us that the rate of change is constant. It doesn’t matter whether I’m going from zero to one mile away from the city centre or I’m going from six to seven miles away from the city centre, that drop in one coordinate is always the same; the rate of change is constant.
So the first thing to say is that the population density is reducing at a constant rate as we travel away from the city centre, but we can be a bit more specific than that. Now that line cut the 𝑦-axis at ten. So when the 𝑥-coordinate was zero, in other words we were at the city centre, the corresponding population density, the 𝑦-coordinate, was ten. So that represents ten thousand people per square mile. So at the city centre, the population density is ten thousand people per square mile.
And as you move away from the city centre, every time you increase the 𝑥-coordinate by one, the population density falls by one on the 𝑦-coordinates. So that represents one thousand people per square mile. So we’ve got four different ways of representing our function, and we can get useful information about the situation and the rates of change from each of those.
Let’s go and have a look at some questions and put our skills to the test then. Number one: which graph could represent the table of values given? So we’ve got a table of values and we’ve got four different graphs. So what we need to do is check each of the graphs to see if those coordinates are on the graph. So extracting the ordered pairs of coordinates from the table of values, zero, six — when 𝑥 is zero, 𝑦 is six — one, three, and two, zero. So on the first one, zero, six, zero is the 𝑥-coordinate and six is the 𝑦-coordinate, so that’s on the graph. One, three, no that’s not on the graph. And two, zero is also not on the graph. So it’s not graph a.
Let’s look at b. zero, six, so that’s this point here. one, three, that’s not quite on the graph. And two, zero is on the graph. Well again, not all of those points from the table values are on the graph, so that’s not the right answer.
For c, we’ve got zero, six, we’ve got one, three, and we’ve got two, zero. All the points are on the graph, so that matches. Let’s just take a look at the last one as well. zero, six, yep that’s on the graph. one, three, yep that’s on the graph. And two, zero is also on the graph, so that would also work. So we got two possible answers: c or d.
Number two: which equation matches the graph? So we’ve got some equations here and we know about the slope and we know about the intercept. So let’s check the intercept first. It cuts here at zero, six. And all of those different equations end with a plus six or just a six. So at the moment, we haven’t ruled anything out as such. But looking at this last equation here, this isn’t in the form 𝑦 equals 𝑚𝑥 plus 𝑏; it doesn’t look like a straight line. It’s- in fact, it’s a cubic; it’s a curvy thing. It’s got 𝑥 cubed; it’s got 𝑥 squared . So this is not gonna be a straight line, so we definitely know this is not the correct answer.
So now we need to look at the slope. Now what we can see is that when I increase my 𝑥-coordinate by one, the 𝑦-coordinate — if I can roll way back down here to the next point on the line — decreases by three. So negative three is the slope of that line, so that’s not this one. This one doesn’t have an 𝑚𝑥 term, so it’s not that one. And this one has got negative three, so that looks like it’s the answer; it matches the slope and it matches the intercept. So that’s our answer.
Now let’s just — before we move on, let’s imagine that this question had been slightly different; and instead of giving us a graph, they’d given us a table of values which had three points on that graph, say this point here, this point here, and this point here. Now we’d have to plug these 𝑥-values into these equations and see if we get the correct corresponding 𝑦-coordinates.
So for the first equation, doesn’t work. For the second equation, well the 𝑦-coordinate’s always six regardless of the 𝑥-coordinates, so that doesn’t work. For the third equation, yep all of those work: zero, six, one, three, and two, zero. So that could be an equation that matches that table of values. And in fact if we work them out, it takes a bit longer for equation d, they all come up correct as well. So when 𝑥 is zero, 𝑦 is six; when 𝑥 is one, 𝑦 is three; when 𝑥 is two, 𝑦 is zero. So that also could be the same equation. So this is a bit weird; we’ve got a straight line and we’ve got a curvy, and they both seem to have some po- some points in common.
Well look, here’s what that curve actually looks like if we plot it out. And you can see that zero, six, one, three, and two, zero are in fact all on the-the curve. So graphs tell you a slightly bigger story than tables of values. If we just use table of values, it’ll look like two possible equations could match them. But when we were looking at the graphs themselves, we were able to be a little bit more discerning about whether or not the graph would match a particular equation because we know a bit from the format of the equation whether it’s a straight line or whether it’s a curvy line.
Now number three, which function table could have been produced by the function of 𝑦 equals two 𝑥 minus eight? Now what we’ve gotta do is we’ve gotta try out tediously all of these different coordinate pairs and see if we get the correct 𝑦 answer for that function.
So looking at a, when 𝑥 is zero, 𝑦 is two times zero minus eight, which is negative eight; when 𝑥 is one, 𝑦 is two times one take away eight, which is negative six; and when 𝑥 is two, 𝑦 is two times two take away eight. That’s four take away eight. So the first two were correct, but the third one wasn’t. So a is not the correct table of values.
And for function table b, we’ve got the same 𝑥-values to plug in. So plugging those into the same equation, we’re getting the same answers negative eight, negative six, and negative four. And none of those match the function table, so b is wrong as well. Let’s hope that c is correct.
Now with c, we’ve got some slightly different 𝑥-values. So they’re not going up in ones they’re going up in twos, but let’s plug those values in as well. So when we plug 𝑥 is negative two into 𝑦 equals two 𝑥 minus eight, two lots of negative two are negative four take away another eight is negative twelve. When we put 𝑥 equal to zero, we got two lots of zero takeaway eight. So nothing take away is negative eight. And putting in 𝑥 equals two, two lots of two are four take away eight is negative four.
And hey presto! They are the correct values, so c is our answer.
And in this question, we could’ve said function table; we could’ve said table of values. In fact, we probably did change between the two as we were talking about it, but they both mean the same thing.
So number four, which graph best matches the following situation? Bill buys a car for twenty thousand dollars and it reduces in value by four thousand dollars a year. Well the first thing we need to do is state an assumption. We have to assume that 𝑥 is the number of years since the purchase was made and that 𝑦 is the value of the car in dollars, so you do need to state that assumption.
So straight away, looking at graph b, it starts off with a value of zero and after five years it’s gone up to a value of twenty thousand dollars. Well that’s not the situation at all; the car is going down in value not up, so b is not correct. So now we’re looking at the two graphs a and c which start off at twenty thousand dollars and go down to zero in value, but it’s a question of how long does it take to do that.
And if I keep taking four thousand away from twenty thousand, I only get to do that five times before it becomes zero, so twenty thousand divided by four thousand is equal to five years. So the car takes five years to depreciate down to zero in value, so a is our answer.
And number five, Arnie the plumber charges for his time according to the graph below. Bob the plumber charges twenty dollars to be called out and then thirty dollars per hour for his time. So we can see here that Arnie doesn’t charge anything for a call out. But one hour, he’s charged forty dollars; after two hours, he would charge eighty dollars and so on.
So the first part of this question is which plumber charges more per hour for their time. So what I’m gonna do is I’m going to plot Bob’s line on the graph as well and we’re gonna compare those two graphs. So Bob charges twenty dollars to be called out. So for no time at all, you’re paying twenty dollars for Bob and then it’s thirty pounds [dollars] an hour for his time. So every time I increase my 𝑥- coordinate by one, I have another hour, I go up by thirty dollars.
Right. Now for Arnie, when I increase my time by one, the rate of pay goes up from zero to forty. So Arnie charges forty dollars for an hour. And from the question, we saw that Bob charges thirty dollars per hour. And forty is bigger than thirty, so Arnie charges more per hour for his time.
Now for part b, if I had the job that took three hours to do, which would be the cheapest plumber? Well three hours, just reading this off from the graph, Bob would be a hundred and ten dollars and Arnie would be a hundred and twenty dollars. So Bob would be cheaper, and well we kinda expected that because Bob is charging less per hour for his work. But look, if I had a job which took no time at all or one hour, then in fact Bob would be more expensive; his line is above Arnie’s line on this, so he would actually be more expensive because of his call out charge that he charges at the very beginning of his visit.
And the third question then in this: I have a job that would cost the same whether Arnie or Bob do it. How long would the job take? Well, when the lines cross over at this point here, they’re both charging the same rate for that job, and that is at two hours. So for two hours, Arnie and Bob both charge eighty dollars.
So when you’re given information in two different forms, so we were given a graph and we were given a verbal description, we’ve got choices: we can either put them all on the same graph or we could convert them both into equations.
And in this case, we’ve- we’d have needed to make the assumption that 𝑦 is the cost in dollars and 𝑥 is the number of hours that they spend on the job. So for Arnie, the intercept would be at zero, so it’s gonna be plus zero on the end. And for Bob, the intercept was at twenty, so we’d have a plus twenty on the end. And the multiplier, so the slope of the line, for Bob we said was thirty. So every time we add an extra hour, the cost goes up by thirty dollars. And for Arnie, the slope was forty. So every time we increase the number of hours by one, the price goes up by forty dollars. So obviously on Arnie’s equation, forty 𝑥 plus zero.
We probably wouldn’t bother writing the plus zero on the end. And Bob’s equation would be 𝑦 equals thirty 𝑥 plus twenty. So we could’ve just run that whole set of questions by putting in the values for 𝑥 and coming out with our numbers that way. How do you do it doesn’t really matter as long as you compare things using the same method, either the graph or the equations.