### Video Transcript

The graph of the linear function ๐ is shown in the ๐ฅ๐ฆ-plane. The slope of the graph of the linear function โ is three times the slope of the graph of ๐. If the graph of โ passes through the point two, one, what is the value of the function โ when ๐ฅ is equal to five?

So if we take a look at the question, the first thing thatโs mentioned is the slope cause weโre told that the slope of the graph of the linear function โ is three times the slope of the graph of ๐. So weโll need to calculate the slope. And how will we do that? Well, we have a formula to help us calculate the slope. That is, ๐ which is our slope is equal to ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one.

So what this means is if we have two points, itโs the change in ๐ฆ between them divided by the change in ๐ฅ or itโs 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 weโve 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 weโve said itโs a line. This means that our slope will remain constant throughout the length of it. And we know that because weโre also told that the graph is a linear function. So that means itโs going to be a straight line. A good tip for when youโre selecting points to use when youโre trying to find the slope is to choose points where itโs easy to read off the ๐ฅ- and ๐ฆ-coordinates because this will make life easier when weโre solving the problem.

So the points that Iโve chosen are zero, two and negative six, zero. And I know that these are the coordinates because weโre told in the question โ well on the graph โ that one square is equal to one unit. So weโve got our two points. Like I said, you could have chosen any two points along the line because it will not affect the slope. Iโve just chosen these two because theyโre easier to read.

So now, what Iโve done is Iโve labelled the coordinates. So we have ๐ฅ one, ๐ฆ one: negative six, zero and ๐ฅ two, ๐ฆ two: zero, two. And then, if we substitute these into our formula for the slope, we can say that the slope of our function ๐ is gonna be equal to two minus zero โ so ๐ฆ two minus ๐ฆ one โ divided by zero minus negative six. Well, this is gonna give us a slope of two over six. And thatโs because two minus zero is two and then zero minus negative six is going to be six because if you subtract a negative, itโs the same as adding. And then we can simplify this by dividing numerator and denominator by two. So itโs gonna give us a third.

So we can say that the slope of the linear function ๐ is going to be a third. Well, how is this useful? Well, itโs useful because we can use this now to work out the slope of our function โ. And thatโs because we were told that the slope of the linear function โ is three times the slope of the graph of ๐. So therefore, the slope of โ is gonna be equal to three multiplied by a third which is just going to be equal to one. So great, weโve now found the slope. But then, how weโre 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 โ is two, one to find the formula or equation of our straight line of the function โ. 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 ๐ฆ is equal to ๐๐ฅ plus ๐, where ๐ โ as we already know โ is the slope and ๐ is our ๐ฆ-intercept. So this is where our line is going to cross the ๐ฆ-axis.

So now what we can do is form the equation for our linear function โ. And the difference is instead of having ๐ฆ, weโre gonna have โ of ๐ฅ. So if we substitute in our value for the slope, we can have โ of ๐ฅ is equal to ๐ฅ or one ๐ฅ because the value of our slope was just one โ we wouldnโt usually write the one, so Iโve put it in a bracket โ plus ๐. So what do we need to find? Well, we need to find out ๐. So what is the ๐ฆ-intercept? Well, to work this out, we can use the point that weโve got because we can substitute in ๐ฅ equals two and ๐ฆ equals one, remembering that one is going to be our โ of ๐ฅ.

So when we do that, we get one is equal to two plus ๐. So then, we subtract two away from each side of the equation. And when we do that, we get negative one is equal to ๐. So we now have our ๐ฆ-intercept. So now if we substitute that back into the general form for the equation of our function โ, we get โ of ๐ฅ is equal to ๐ฅ minus one. So what we need to do now to solve the problem is find out the value of our function of โ when ๐ฅ is equal to five. So what weโre going to do is substitute in ๐ฅ equals five. So when we do that, we get five minus one. And thatโs because ๐ฅ now becomes five.

So therefore, we can say that the value of our function โ when ๐ฅ is equal to five is going to be four.