A linear function is a function of the form F of X equals a X Plus B where A&B represent real numbers. And when we show this graphically, a represents the gradients of the function and B represents the Y axis intersect, which is sometimes called the vertical intercept. Now what do you think would happen if we varied a? Well, let's have a look at a few examples. Because we're looking at the graphs of linear functions, that means we're going to be looking at straight lines, and so plot a straight line. We only need two points, however, we often choose three points because the Third Point is a good check to make sure we haven't made a mistake, so let's have a look at F of X equals X +2. OK, first points I look at is F of 0. Now F of zero 0 + 2, which is simply too. S is one. Is 1 + 2, which gives us 3. An F of two. 2 + 2 which will give us 4. OK, for the next function, let's look at F of X equals 2 X +2. F of X equals 2 X +2. So we get F of 0. Equals 2 * 0, which is 0 + 2, which gives us 2. S is one which gives us 2 * 1 which is 2 + 2, which gives us 4. And F of two which gives us 2 * 2, which is 4 + 2, which gives us 6. There's no reason why I shouldn't be negative, so let's look a few negative values. If we had F of X equals minus two X +2. We would have FO equals. Minus 2 * 0 which is 0 + 2, which gives us 2. F of one which gives us minus 2 * 1. Which is minus 2 + 2, which gives us 0. An F of two which gives us minus 2 * 2 which is minus 4 + 2 which gives us minus two. And finally we'll look at F of X. Equals minus X +2. So we've got F of 0. Equals 0 + 2, which is 2. S is one which equals minus 1 + 2, which equals 1. And finally F of two which is minus 2 + 2 which equals 0. Now what we're interested in doing is looking at the graphs of these functions. So if we have our axes drawn with F of X on the vertical scale an X on the horizontal axis, the first function we looked at was F of X equals X +2, which gave us points at 02. Second point resort. 13 Our third points was at 2 full. And when we join this up, we expect a straight line. We can label less. F of X. Equals X +2. The second function we looked up was F of X equals 2 X +2. Which games, the points 02, which we've already marked here, was the .1 four. And it gave us the .2. 6. We should be able to draw these with a straight line. We can label SF of X. Equals 2 X +2. The next function we looked up was F of X equals minus two X +2, and once again this gave us a points at 02 appoint at one zero and a point at two and minus 2. And when we join these up as before, we expect a straight line. We can label less. F of X. Equals minus two X +2 and the final function we looked at was F of X equals minus X +2 and this gave us a point at 02 again point at one one. Anna points AT20. We can join those up to get a straight line. This is F of X. Equals minus X plus so. Now first thing we notice about these graphs is that they all crossed 2 on the F of X axis. That's be'cause be value is 2 in every single function and be represents the Y axis intercept. What we were interested in is what happens as the value of a changes. Now when A is positive, the line goes up and the bigger the value of A, the faster the line goes up as X increases. And when A is negative, the line goes down. And the bigger the value of an absolute terms, the faster the line goes down as X increases. OK, So what happens as we very be? Well, that's always good place to start is by actually looking at few examples. So let's consider the example F of X equals 2X plus 3. F of 0 here would be 2 * 0 + 3, which is 0 + 3, which is just three. F of one is 2 * 1, which gives Me 2. Plus three, which gives me 5. An F of two. Gives Me 2 * 2 which is 4. Plus three, which gives me 7. OK, Next One next functional look at is F of X equals 2X plus one. OK, for this function we get F of 0 is equal to 2 * 0, which is 0 plus one, which gives me one. I have one gives Me 2 * 1 which is 2 plus one which gives me 3. And F of two gives Me 2 * 2, which is 4 + 1, which gives me 5. And the final function I want to look at is F of X equals 2X minus three. F of X equals 2X minus three, so F of 0. Is 2 times here, which is zero takeaway 3 which is minus 3. F of one. 2 * 1 which is 2 takeaway. Three gives me minus one and finally F of two. Which is 2 * 2, which is 4 takeaway three, which gives me one. So what we're interested in doing is looking at the graphs of these functions. So as usual, we have RF of X on the vertical axis and X one horizontal axis. So first function we talked about was F of X equals 2X plus three and the points we had were zero and three. 15 And two. And Seven. We can join those up. With a straight line label up F of X equals 2X plus 3. The next function we looked up was F of X equals 2X plus one and the points we had there were. Zero and one. One and three. Two and five. Once again, we can draw those. Join those up with a ruler. Label at one F of X. Equals 2X plus one. And the final function looked up was F of X equals 2X minus three. And the points we had were 0 - 3. One and minus one. And two. And warm. But enjoying those off. As before. With a ruler. We label list we get F of X equals 2X minus three. OK, first thing we notice here is that all the graphs are parallel. In fact they have the same gradients, and that's because in each case the value of a was two. So all the graphs have a gradient of two and we also notice that as we varied B, when B was three. The graph of the function went through three on the F of X axis. Would be was one the graph of the function went through one on the F of X axis and when be was minus three. The graph of the function went through minus three on the F of X axis. OK, so we know what happens when I'm being positive and when A&B are negative. What happens if A&BRO? Well, let's see what think about what happens when a equals 0 first of all. So if A equals 0 we get a function of the form F of X equals a constant, so that could be for example, F of X equals 2. Or F of X equals minus three. Just a couple of examples. We can sketch what they might look like. F of X axis here. Now X axis here F of X equals 2. That means for. Whatever the value of X, the F of X values always two. So in fact we just get a horizontal line. Which comes through two on the F of X axis. So if of X equals 2. And when F of X equals minus three, we get a horizontal line. That just comes through. Minus three on F of X axis. So that's what happens when a equals 0. What about when B equals 0? But let's have a look. The B equals 0. We get a function of the form F of X equals a X and as we said at the beginning, a can be any real number. So, for example, we might have F of X equals 2X or F of X equals minus 3X. OK, and as we've already said, what happens when we use? These values of AF of X&X. For looking at F of X equals 2X. It's going to come through the origin because B equals 0, so it will cross F of X at 0. And it will have a gradient of two since a IS2. So it's a sketch. This could represent F of X equals 2X. Of X equals minus three X once again will go through the origin because B equals 0. And it has a gradients of minus three. Remember the minus means the line is coming down and the three means that it's going to be a bit steeper than it was before, so it might be like this. F of X equals minus 3X. OK, Lastly I want to look at functions which are not in the form F of X equals a X plus B. So. What would we do? So we want our functions in form F of X equals X plus B. It's quite useful, so you can think about now if we used Y equals F of X just for convenience. So suppose I had. 4X minus three Y. Equals 2. First thing we want to do is make Y the subject of this equation. So if I had three Y answer both sides 4X equals 2 + 3 Y. Now I want to get three wide by itself, so I need to take away 2 from both sides. So over here I got 4X takeaway 2 on this side. If I take away too, we just get left with three Y. And so finally to make why the subject I need to divide both sides by three. So we get 4 thirds of X. Minus 2/3 equals Y. And as we said before, Y equals F of X. So this means our function is actually F of X equals 4 thirds X minus 2/3. So this function represents a straight line with the gradients of Four Thirds and Y axis intercept of minus 2/3. What about if we had two X minus 8 Y plus eight Y minus one equals 0? Once again, we want to make why the subject of the equation so a natural first step would be to add 1 to both sides. So 2X plus eight Y equals 1. Next thing you want to do to get 8. Why by itself is to subtract 2 X from both sides. If we subtract 2 actually miss side, we just get left with a Y and this side we get one takeaway 2X. And finally we need to divide both sides by eight since we just want why we've got eight, why there? So divide both sides by it. We got Y equals 1/8 - 2 over 8 times X and obviously ones are functioning to form a X Plus B, which means we would change around. Just rearrange this right son side here to get Y equals minus 2 eighths of X plus 1/8. And we can simplify minus 2 eighths to be minus 1/4. So we get minus one quarter of X Plus one 8th. And as we said before, Y equals F of X. So here we have it. We are function is F of X equals minus one quarter X Plus one 8th, and graphically this is represented by a straight line with the gradients of minus 1/4 and yx intercept of 1/8. What about if we have this example? Y equals. 13 X minus 8. All divided by 5. Now a little why is already the subject of the formula. It's not quite in the required form, and that's because of this divide by 5. But we can just rewrite the right hand side as Y equals 13 X divided by 5 - 8 / 5 and since why is F of X? We can write this as F of X equals 13 over 5X. Minus 8 over 5. So this function is represented graphically by a straight line with the gradients of 13 over 5 and a Y axis intercept of minus 8 fifths.