In this video, we’re gonna use our skills and knowledge of linear functions, tables of values, graphing, and solving equations to construct equations that model linear relationships. We’ll also use the model to analyse the relationship. For example, examining initial conditions and rates of change, interpreting the significance of the values of the coefficients in the equations and the constants. Now if you’re new to this topic and all of that technical language didn’t make much sense, what we’re gonna be doing is looking at some real-world scenarios and making equations to fit them.
Right! Let’s look at our first example. A gardener is adding plant food to water to put on their plants to help them grow stronger. The instructions say to add two ounces of plant food to every three gallons of water. So every three gallon watering can that they pour on their plants has got three gallons of water and two ounces of plant food. So what we’re gonna do is we’re gonna come up with an equation that describes what’s going on here, and then we’re going to draw a graph. And then we’re gonna sort of discuss the interpretation of the different bits of the equation and-and what the graph looks like.
Now first, we need to define some variables. So we’re gonna be letting 𝑥 equal the number of gallons of water we’re going to use to do our watering and 𝑦 be the number of ounces of plant food that we’re gonna put into that water before we water our plants. And we’re gonna produce an equation in the format 𝑦 equals 𝑚𝑥 plus 𝑏. We reckon there’s a linear relationship between these two things as we add more gallons of water. We’re going to add more ounces of plant food at the same rate for every gallon of water. So it makes a straight line graph; 𝑦 equals something times 𝑥 plus another number. And the multiplier of 𝑥 is the slope of the graph. So every time I increase my 𝑥-coordinate by one, how much is the 𝑦-coordinate gonna go up by?
And this other value, this number that we’re adding on at the end here, that is the 𝑦-intercept where it cuts the 𝑦-axis. So if I had an 𝑥-coordinate of zero, this first term will be whatever ever the slope is times zero. So that’s gonna be zero plus this number. So that’s the 𝑦-coordinate when the 𝑥-coordinate is zero.
Now we know that when 𝑥 is three when we’ve got three gallons of water, then 𝑦 is two. We’re using two ounces of plant food. Now given that the slope is when I increase 𝑥 by one by how much does the 𝑦-coordinate change, what I’d like to do is see how many ounces of plant food I’m gonna use when I use one gallon of water. Now we know that when we use three gallons of water, we get two ounces of plant food. So if I have a third as much water, if I divide that by three, I’m working at one gallon of water. So if I’ve got a third as much water, I’m gonna use a third as much plant food. So I need to divide two by three. Now two by three is not a nice easy number, if you try and represent it as a decimal. But if we just represent that as a fraction, as a rational number, we can just write two over three; two divided by three. So two-thirds is gonna be the slope.
Now also from the description that we’ve got in the question, if we have no water, we’re not gonna add any plant food to that. So we’ve got there the 𝑦-intercept. When 𝑥 is zero, we’ve got no water, then 𝑦 equals zero. We’ve got no plant food, so there’s our 𝑦-intercept. So I can put those into my equation: 𝑦 is two-thirds of 𝑥, because that’s the slope, plus zero, because that’s the intercept. Now clearly, we don’t have equations with plus zero on the end. We don’t actually need to write that. So let’s just get rid of that, and our equation becomes 𝑦 just equals two-thirds of 𝑥.
So now let’s go ahead and sketch a graph of that. Now putting in the points on this graph is a little bit tricky. Because every time I increase my 𝑥-coordinate by one, I’ve gotta go up by two-thirds. And that’s quite difficult to judge on-on this grid. But I do know that if every time I go across three, so I have three gallons, then I go up two. So for three gallons, I’ve got two ounces. So three more gallons, I’ve got two more ounces. For three more gallons, I’ve got two more ounces and so on. So for zero, it’s zero, and then I can go backwards, minus three. I go down, two minus another three. I go down another two minus another three and go down another two. Now I can join all those together. And that’s what our graph would look like.
Now the graph cuts the 𝑦-axis at zero. And that means when we’ve got zero gallons of water, we’re adding zero ounces of plant food. So that’s the interpretation of that. The slope of the line every time I at one to my 𝑥-coordinate I go up two-thirds means every time I want to use another gallon of water, I have to add two-thirds of an ounce of plant food. So that’s that or is it? Well wait a minute. It doesn’t really make much sense, does it? We-we can’t pour on negative gallons of water onto our plants, so we have to restrict this function. We have to restrict the domain of this function for 𝑥 to be greater than or equal to zero. It just doesn’t make sense to have negative amounts of water. So because of the real-life constraints that we’re putting on this function, we’re saying the function only works when 𝑥 is bigger than or equal to zero. We’re not gonna deal with negative amounts of water.
Now in our second question, Sheena is gonna decorate some rooms using paint that takes three gallons to cover two hundred and ten square feet. Construct a function to represent the relationship between the number of square feet to be painted and the number of gallons of paint required. Now the question seems to suggest that we’re controlling the number of square feet to be painted, so that’s gonna be our independent variable 𝑥. And the number that pops out from our equation is gonna tell us how many gallons of paint we require, that’s gonna be the dependent variable 𝑦.
Now we also know from the question that two hundred and ten square feet of wall requires three gallons to cover it. But to construct my equation, I need to know the slope. So I need to know what each extra one 𝑥 variable, so where each extra one square foot of wall, how many gallons that would require? So instead of two hundred and ten square feet, I want to know what one square feet requires. And if I divide two hundred and ten by itself by two hundred and ten, that gives me one square foot. So if I’ve got one two hundred and tenth as many square feet, I’m gonna have one two hundred and tenths as many gallons to paint it. So each square foot requires three two hundred and tenths of a gallon, which simplifies to one-seventieth of a gallon of paint in order to paint it. So the slope of our line is gonna be one-seventieth.
And the question implies that if we’ve got no wall, then we don’t need any paint. So the 𝑦-intercept is zero. In other words, when 𝑥 equals zero, the 𝑦-coordinate is array. It cuts the 𝑦-axis at zero. So with those two bits of information, I can write down my equation: 𝑦 equals the slope times 𝑥, one-seventieth of 𝑥, plus the 𝑦-intercept, plus zero. And of course we don’t normally go around writing plus zero on the end of our equations, so we can simplify it down to this: 𝑦 equals one-seventieth of 𝑥. So there’s our equation, our function.
Now if we were to sketch that graph, this is what it would look like. It will be a very, very, very shallow line. We’d have to go along seventy, along the 𝑥-axis, in order to go up one on the 𝑦-axis. So the scales, they’re very different on those two things. But again like last time, we need to think very carefully about the domain of this. It doesn’t make sense to pay-to paint negative amounts of wall. So we’re gonna restrict the domain of our 𝑥-axis, the-the inputs that we can have to our function, to be greater than or equal to zero. So because of the constraints of the real world, our function is only defined from zero upwards. We’re not gonna deal with negative areas and negative amounts of paint.
Now in this last example, slightly more interesting example in some ways. Ringring smartphones charge a one-off fee of fifty dollars for a cell phone, and then ten dollars a month for calls and data. Now we’re gonna do this question in one way, and then we’re gonna look at a bit more closely and see that actually needs to be done in a slightly different way. So don’t start screaming out loud if you don’t agree with what we’re doing to start off with. So we’re gonna define a couple of variables; 𝑥 the independent variable, the thing that’s gonna change, is the number of months. And then that’s gonna determine what the total cost of the deal is, so 𝑦 is the dependent variable. That’s gonna be the cost in dollars. Now when 𝑥 is zero, at the beginning of the deal, you just pay the fifty dollars for the phone. So the 𝑦-intercept is gonna be fifty. When the 𝑥-coordinate is zero, the 𝑦-coordinate is fifty.
And increasing 𝑥 by one, or adding one month to our deal, puts up the cost by ten dollars. So every time we had an extra month, the cost goes up by ten, that means the slope is ten. So we can quickly write down our equation: 𝑦 is equal to ten 𝑥, the slope times 𝑥, plus fifty, the start-up cost of the deal. And straight away, we can see that this doesn’t make any sense to have negative 𝑥 values again, so we can restrict the domain of our function. It only makes sense for 𝑥-values to be greater than or possibly equal to zero, so that’s our domain.
So if we wanted to plot this function, so we can see the graph when 𝑥 is zero, 𝑦 is fifty. When 𝑥 is one month, then we’ve got add ten to that is sixty. When 𝑥 is two months, we add another ten dollars and so on. So we keep adding ten dollars every month, our graph looks like this. And that matches both the real-life situation because we’ve got a start-up cost of fifty dollars. And every time we increase the number of months by one, the corresponding cost goes up by ten. And those things also match the equation that we created.
So that’s great, but even that isn’t quite right. Because in reality, the company will probably charge you at the start of each month and you wouldn’t get any refund if you cancel the contract at any point in time. So this idea that if you did three and a half months, it would cost you eighty-five dollars doesn’t really work in reality. The truth is it would be more like a step function. So this is probably what the cost would really look like in real life. And this isn’t a function in the sense that we’ve been looking at our functions. It’s not just a straight line linear function. You can see that we-we call it a step function, because the prices step up as we go through the months. So the reality is they’d probably charge you the fifty dollars one off fee, and also the first month’s phone usage of ten dollars. So straight away, your cost is gonna be sixty dollars for the first month. Then all of the sudden at the beginning of the second month, they’re gonna charge you for the second month’s calls and your cost jumps up to seventy dollars. And you know whether you use your phone for one day that month or for the whole of that month, the cost is still seventy dollars.
So we could have modeled this as a linear function which we did, which is kind of an approximation of what’s going on. Whereas in reality, the real function will probably look more like this. Now step functions aren’t really in the curriculum at the moment. That’s not really what we’re supposed to be learning about, but we just wanted to give you a bit of insight into another way of modeling real life, perhaps a little bit more accurately.
So let’s summarise what we’ve just been looking at then. So with these functions and the real-life situations, first of all we need to define our 𝑥 and 𝑦 variables. So 𝑥 is the independent thing, the thing that you’re varying, and 𝑦 is the output from your function, the thing that depends on the 𝑥-value. Then we need to calculate the rate of change. This is gonna be the multiplier of 𝑥 in our function equation, and it tells you the slope of the line when we draw the graph. And thirdly, work out the value of 𝑦 when 𝑥 is equal to zero. So this is the 𝑦-intercept or it’s where the graph cuts the 𝑦-axis.
And actually you need to think about the domain. Does it make sense in the real world to have all values going into that function. We saw we didn’t wanna have negative amounts of area to paint. We didn’t wanna have negative amounts of gardening to do. And next we need to write the equation. Now if that’s a linear function, then it’ll be in the form of 𝑦 is equal to 𝑚𝑥 plus 𝑏. 𝑦 is the slope times the 𝑥 variable plus the 𝑦-intercept, the 𝑦-coordinate where it cuts the 𝑦-axis. And finally, you can go on to draw the graph.