The graph of the linear function 𝑓 is shown in the 𝑥𝑦-plane. The slope of the graph of the linear function ℎ is three times the slope of the graph of 𝑓. If the graph of ℎ passes through the point two, one, what is the value of the function ℎ when 𝑥 is equal to five?
So if we take a look at the question, the first thing that’s mentioned is the slope cause we’re told that the slope of the graph of the linear function ℎ is three times the slope of the graph of 𝑓. So we’ll need to calculate the slope. And how will we do that? Well, we have a formula to help us calculate the slope. That is, 𝑚 which is our slope is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.
So what this means is if we have two points, it’s the change in 𝑦 between them divided by the change in 𝑥 or it’s sometimes known as the rise over the run. So the first thing that we want to do is we want to work out the slope of the line that we have on our graph. So we’ve got the formula. But how are we going to use it? Well, what we want to do is select two points on our line.
It is worth mentioning that as we’ve said it’s a line. This means that our slope will remain constant throughout the length of it. And we know that because we’re also told that the graph is a linear function. So that means it’s going to be a straight line. A good tip for when you’re selecting points to use when you’re trying to find the slope is to choose points where it’s easy to read off the 𝑥- and 𝑦-coordinates because this will make life easier when we’re solving the problem.
So the points that I’ve chosen are zero, two and negative six, zero. And I know that these are the coordinates because we’re told in the question — well on the graph — that one square is equal to one unit. So we’ve got our two points. Like I said, you could have chosen any two points along the line because it will not affect the slope. I’ve just chosen these two because they’re easier to read.
So now, what I’ve done is I’ve labelled the coordinates. So we have 𝑥 one, 𝑦 one: negative six, zero and 𝑥 two, 𝑦 two: zero, two. And then, if we substitute these into our formula for the slope, we can say that the slope of our function 𝑓 is gonna be equal to two minus zero — so 𝑦 two minus 𝑦 one — divided by zero minus negative six. Well, this is gonna give us a slope of two over six. And that’s because two minus zero is two and then zero minus negative six is going to be six because if you subtract a negative, it’s the same as adding. And then we can simplify this by dividing numerator and denominator by two. So it’s gonna give us a third.
So we can say that the slope of the linear function 𝑓 is going to be a third. Well, how is this useful? Well, it’s useful because we can use this now to work out the slope of our function ℎ. And that’s because we were told that the slope of the linear function ℎ is three times the slope of the graph of 𝑓. So therefore, the slope of ℎ is gonna be equal to three multiplied by a third which is just going to be equal to one. So great, we’ve now found the slope. But then, how we’re going to use this to help us?
Well, we can use the slope and the fact that we know that a point on the graph of ℎ is two, one to find the formula or equation of our straight line of the function ℎ. And we can do that using the general form for the equation of a straight line, so the equation of a linear function. And that is, that 𝑦 is equal to 𝑚𝑥 plus 𝑏, where 𝑚 — as we already know — is the slope and 𝑏 is our 𝑦-intercept. So this is where our line is going to cross the 𝑦-axis.
So now what we can do is form the equation for our linear function ℎ. And the difference is instead of having 𝑦, we’re gonna have ℎ of 𝑥. So if we substitute in our value for the slope, we can have ℎ of 𝑥 is equal to 𝑥 or one 𝑥 because the value of our slope was just one — we wouldn’t usually write the one, so I’ve put it in a bracket — plus 𝑏. So what do we need to find? Well, we need to find out 𝑏. So what is the 𝑦-intercept? Well, to work this out, we can use the point that we’ve got because we can substitute in 𝑥 equals two and 𝑦 equals one, remembering that one is going to be our ℎ of 𝑥.
So when we do that, we get one is equal to two plus 𝑏. So then, we subtract two away from each side of the equation. And when we do that, we get negative one is equal to 𝑏. So we now have our 𝑦-intercept. So now if we substitute that back into the general form for the equation of our function ℎ, we get ℎ of 𝑥 is equal to 𝑥 minus one. So what we need to do now to solve the problem is find out the value of our function of ℎ when 𝑥 is equal to five. So what we’re going to do is substitute in 𝑥 equals five. So when we do that, we get five minus one. And that’s because 𝑥 now becomes five.
So therefore, we can say that the value of our function ℎ when 𝑥 is equal to five is going to be four.