Video Transcript
In this video we’re gonna see how to represent the same function in different ways. We’ll be using algebra, graphs, and tables of values and also verbal descriptions. And we’ll be looking for links between them.
Let’s start with this example. Tonya lives at the bottom of a hill, and her friend, Herb, lives at the top of the hill. It’s always one degree cooler at the top of the hill than it is at the bottom. Let 𝑥 represent the temperature where Tonya lives and 𝑦 represent the temperature where Herb lives. So here is a quick sketch to represent that. We’ve got Tonya’s place down the bottom of the hill and Herb’s place at the top of the hill. Now we’re gonna assume that the temperatures are measured in Celsius. So Tonya’s place is 𝑥 degrees Celsius and Herb’s place is 𝑦 degrees Celsius. So sketching a little diagram can help us to understand what’s happening. Now to calculate the temperature when Tonya lives, we just add one degree to the temperature where Herb lives. Or to put it in other way to calculate the temperature where Herb lives, we just subtract one from the temperature where Tonya lives. Now we can create an algebraic expression. The temperature where Herb lives we’ve called 𝑦 and that is one less than the temperature where Tonya lives, which we’ve called 𝑥. So 𝑦 is equal to 𝑥 minus one. This is the algebraic expression that describes the relationship between the temperatures at Tonya’s and Herb’s places.
We can also make a table of values for a few specific temperatures. The best idea here is to pick some 𝑥-values at equal intervals. This means we can get a sense of the rates of change. In other words as we increase the 𝑥-coordinate by a certain amount, what happens to the corresponding 𝑦-coordinates? Do they change at a constant rate? Do they change increasingly? Do they increase or do they decrease? Now this might not be typical of all the temperatures that Tonya and Herb are gonna experience in life. But I’m just gonna pick these values here. So the 𝑥-values, which is Tonya’s temperatures, might be minus two, minus one, zero, one, and two. And these increase one at a time. So that’s quite a good idea to have equal intervals between our 𝑥-coordinates. And we can see from our algebraic expression over here that the 𝑦-coordinate, the temperature where Herb lives, is gonna be one less than the 𝑥-coordinate, the temperature where Tonya lives. So that I’ve worked out the 𝑦-values; they’re one less than the 𝑥-values. Now we can see also that every time I increase the 𝑥-coordinate by one, the 𝑦-coordinate also goes up by one. So if the temperature moves from say zero degrees up to one degree and down at Tonya’s, then the temperature of Herb’s is gonna be from minus one degree up to zero degrees. That’s also an increase of one degree. So whether the temperature at Tonya’s increases from minus thirty to minus twenty-nine or from one hundred to a hundred and one degrees if it goes up by one degree, then the temperature at Herb’s would also go up by one degree.
And from the table of values, we can also plot the points and spot that they suggest a straight line pattern or a linear relationship between 𝑥 and 𝑦 and join the dots. Well let’s have a go with that. When 𝑥 is minus two, 𝑦 is minus three, so that’s a point on our graph. When 𝑥 is minus one, 𝑦 is minus two, so that’s another point. When 𝑥 is zero, 𝑦 is negative one; when 𝑥 is one, 𝑦 is zero; and when 𝑥 is two, 𝑦 is one. So as we said those points suggest a straight line. Another way we set up our rule that the temperature at the top of the hill is always one degree less than the temperature at the bottom of the hill. This pattern is gonna carry on forever. We’re not saying anything about what the actual temperatures might be. But whatever the temperature of the bottom of the hill, the temperature of the top of the hill will always be one degree cooler. So that pattern is gonna continue as 𝑥 increases or as 𝑥 decreases like this.
So now we’ve represented the same relationship four different ways. We had a verbal description and from that verbal description we were able to create the algebraic expression. But it was important to define the variable so that we’d known what 𝑥 and 𝑦 represent. If we look at the function table or table of values, we can see that for each pair of coordinates the 𝑦-value is one lower than the 𝑥-value. Now this matches the algebraic equation and the description. Whatever the 𝑥-value we use, the corresponding 𝑦-value is one lower. If we look at the graph and read the coordinates of any point on the line, for example, two, one, we find that the 𝑦-coordinate is one smaller than the 𝑥-coordinate, or one, zero, or minus two, minus three.
Now we can also look at the rates of change. The description says that Herb’s place is always one degree cooler than Tonya’s. So if I make Tonya’s place twenty degrees hotter, then Herb’s would also get twenty degrees hotter because the difference in temperature is always the same, one degree. And as we said before, this means that whenever I increase my 𝑥-coordinate by — so if I increase it by one, the 𝑦-coordinate would also increase by one. But if I increase my 𝑥-coordinate by three, so from minus two up to positive one, then the 𝑦-coordinate, the corresponding 𝑦-coordinate, would also increase by three. And this pattern is also visible on the graph. So if we take this point here, the 𝑥-coordinate is negative two. So Tonya’s temperature is negative two. And the 𝑦-coordinate is negative three. So Herb’s temperature is negative three. So if I increase Tonya’s temperature by one degree, then Herb’s temperature also increases by one. So we’ve moved from this point here, where Tonya’s temperature is negative two and Herb’s is negative three up to this point here, where Tonya’s temperature is negative one and Herb’s is negative two.
But let’s say we started off from the same starting position of Tonya’s temperature of negative two, Herb’s being negative three. And then we increase Tonya’s temperature by three degrees up to positive one. Then the corresponding Herb’s temperature has also gone up by three degrees. So regardless of our starting point, regardless of how much we move, the 𝑥-coordinates and the 𝑦-coordinates are always changing at the same rate. We’ve got a constant slope or rate of change.
Now we can also start to make some comparisons between the equation and the graph. So in the equation when 𝑥 is equal to zero, then the corresponding 𝑦-coordinate will be zero minus one, which is negative one. So we’re talking about this point here on our graph; that’s on the 𝑦-axis. So in an equation like this 𝑦 is equal to something times 𝑥 plus or minus some other number, putting in an 𝑥-value of zero is always gonna make this term zero because zero times anything is zero. So the number that we’re left with will tell us where we’re cutting the 𝑦-axis. So this number here without actually having to do any calculations involving zero as we did here — that number here will always tell us where our line is cutting the 𝑦-axis. Now we also know that this expression means 𝑦 equals one times 𝑥 minus one. It’s just that we don’t normally bother writing that first one in. Now this term is always minus one. So if we look at this term here, if I increase my 𝑥-coordinate by one, that first term becomes one times 𝑥 plus one. And when I use the distributive law to multiply that out, so that’s our original number one 𝑥 and we’ve added one to it, this means that by adding one to the 𝑥-coordinate, the 𝑦-coordinate also gets one added.
Looking at another example, if our original equation was 𝑦 equals two 𝑥 minus one and I added one to 𝑥, then it’ll be two lots of the 𝑥-coordinate plus one. And then I could use the distributive law on that to get two 𝑥 plus two. So I would have my original two 𝑥 and I would have added two to it. So now by adding one to the 𝑥-coordinate, the 𝑦-coordinate gets two added. So the thing is whenever I’ve got my equation in this format 𝑦 is equal to something times 𝑥 plus a number, that multiplier of 𝑥 tells us about the rate of change. Every time I add one to my 𝑥-coordinate — every time I try to move along the 𝑥-axis by one — how much is my 𝑦-coordinate gonna change by. Now that multiplier tells us how much the 𝑦-coordinate is gonna change by. If the multiplier was one, the 𝑦-coordinate increased by one. If the multiplier was two, the 𝑦-coordinate increased by two. It tells us about the slope or the steepness of that line. Every time I move across one, how far up am I moving? Am I moving up one? Am I moving up two, up three or whatever? So it’s telling us how steep that line is going to be.
So let’s just look at quick-three quick examples then. 𝑦 equals one 𝑥 plus three, 𝑦 equals two 𝑥 plus four, and 𝑦 equals three 𝑥 plus one. So in each case, if I move just a bit to the right on the graph if I increase my 𝑥-coordinate on by one, what’s gonna happen to the 𝑦-coordinate? Well in the first case when the multiplier was one, the 𝑦-coordinate will increase by one. In the second case when the multiplier was two, it would increase by two. And in the third case when the multiplier was three, it would increase by three. So in the first case, I’ve got a line that goes up like that across one up one and goes up forty-five degrees. In the second case, increasing the 𝑥-coordinate by one increases the 𝑦-coordinate by two. So that’s gonna be a steeper line; I got a long one and up two. And in the third case, it’s steeper still. So this number here in front of the 𝑥, the multiplier of the 𝑥, is telling you about the steepness or the slope of that line.
Let’s look at another example. Connor is saving fifty dollars a week to pay for his holiday. He’s got five hundred and eighty dollars to start with. And his holiday costs nine hundred dollars. Now we’re gonna draw a table of values, we’re gonna draw a graph, and we’re gonna come up with an algebraic equation to represent this. So first of all we need to define some variables. So let 𝑥 be the number of weeks from today, so that’s the time variable, and let 𝑦 equal the total amount of money that Connor has saved in dollars. So let’s start off by creating an algebraic expression. We want to work out an expression that tells us how much money Connor has saved. Well he starts off with five hundred and eighty dollars and every one week — every time we increase 𝑥 by one — he adds another fifty dollars to that. So the multiplier of 𝑥 is gonna be fifty. In fact just to keep things in the same format 𝑦 is equal to 𝑚𝑥 plus 𝑏, 𝑦 is something times 𝑥 plus another number, I’m gonna swap the five hundred and eighty and the fifty 𝑥 around. And you don’t have to swap those two terms around. But if you get into the habit of putting the 𝑥 term first, then you’re less likely to make mistakes because you get used to this pattern. So when 𝑥 is zero, that term, that first term, becomes fifty times zero which is just zero. So we’ve got zero plus five hundred and eighty. In other words, when 𝑥 is zero times zero, in other words today 𝑦 is just equal to five hundred and eighty and that tallies what we knew in the question. And in one week’s time when 𝑥 is equal to one, then we’re gonna have fifty times one plus the five hundred and eighty, which just gonna be six hundred and thirty dollars. So we’ve already started to create a table of values or our function table. At times zero, they’ve got five hundred and eighty dollars. After one week, they got six hundred and thirty dollars. So we can carry on with the same process to fill out the rest of that table of values. After two weeks, they’ll have fifty times two plus five hundred and eighty. After three weeks, they’ll have fifty times three plus five hundred and eighty and so on. So after seven weeks, they’ll have nine hundred and thirty dollars. And now we have enough to go away on that holiday.
Now we can also have a quick look at that table of values and look for the rates of change. So every time I increase my 𝑥-value by one in this table, we can see that the corresponding 𝑦-coordinate increases by fifty. So that tallies with the fifty 𝑥 being the slope or the gradient of that line. The rate of change every time we had one, the 𝑦-coordinate goes up by fifty. Now I can plot those values onto a graph. And plotting zero, five hundred and eighty and one, six hundred and thirty, and all those coordinates, I get these values, which I can actually just join with a straight line. Now in this case I haven’t extended the line off into the future or off into the past because I don’t know how he got his five hundred and eighty dollars, whether he saved at that steady rate to get that, probably not. Because if he started at nothing, he wouldn’t have ended up with five hundred and eighty dollars. And once he’s reached his target in the future, I don’t think he will save fifty dollars a week any more because he’s got enough money for his holiday. So within the time scale from zero to seven weeks, we know that this graph follows a linear pattern. And we can see that every time I increase my 𝑥-coordinate by one, so from zero weeks to one week my coordinates, and my 𝑦-coordinate is increasing by fifty. If I, look at this one here — if I increase this 𝑥-coordinate by one from five to six, then my 𝑦-coordinate, our corresponding 𝑦-coordinate, also goes up by fifty. So this increment is always the same no matter whereabouts on that graph we start. So when I increase my 𝑥-coordinate by one, the corresponding 𝑦-coordinate will go up by fifty.
Okay let’s have a look at one last example to see how we can put on all our knowledge together. Here is a graph showing the depth of water 𝑦 feet at time 𝑥 minutes. The water is draining from the tank. a) Describe the rate at which the water is draining. b) Make a table of values for the depth of the water. And c) write an algebraic function for the depth of the water in the tank. So looking at the graph, it’s a straight line graph. It’s draining at a constant rate. So every time I increase my 𝑥-coordinate by one, in this case the 𝑦-coordinate is going down by the same amount. Increase the 𝑥-coordinate by one, the 𝑦-coordinate is going down by the same amount. So looking at this over the course of the ten minutes here, the water drains from a hundred feet till to nothing. So if the water drains a hundred feet in ten minutes, we could quantify the rate of drainage per minute. And how do I turn ten minutes into one minute? What do I have to divide by? So one minute is a tenth of the amount of time of ten minutes. So if it’s draining at a constant rate and I’ve got a tenth the amount of time, it’s only gonna be a tenth the amount of drainage in one minute. So the drain rate is ten feet for every one minute, so ten feet per minute. So this is our answer to part a). It drains at a constant rate of ten feet per minute.
Now we’re doing a table of values. It’s always good to have a fixed interval — in this case stepping in ones for the 𝑥-coordinate. So we’re stepping up here in one minute, so going zero minutes, one minute, two minutes, three minutes, and so on. Now if we’ve read off the corresponding 𝑦-values, we can fill those in our table of values and there we go. We had to split the table into two separate tables in fact because we didn’t have quite enough room on the page. But we can also see that every time we increase the 𝑥-coordinate by one, the 𝑦-coordinate in this case is decreasing by ten. So if we were creating an equation, that’s gonna give us a clue as to how fast the line is going down — how fast the 𝑦-coordinate is decreasing.
So moving on to part c), the algebraic function we can see it’s a straight line function. And we’ve already discussed that straight line functions are always in this format 𝑦 is something times 𝑥, just a number times 𝑥 plus another number, so we need to try to work out what these numbers are going to be. Well it cuts 𝑦-axis at one hundred. In other words, when the 𝑥-coordinate is zero, this term here is gonna become whatever that number is times zero. So that’s gonna be zero plus a number, so this number here must be a hundred. And we saw that every time we increase our 𝑥-coordinate by one, the 𝑦-coordinate went down by ten. So that is gonna be the multiplier 𝑚. So our function is 𝑦 is negative ten 𝑥 plus a hundred.
