Plot Linear Equations
We know that every single linear equation is in the form 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the gradient and 𝑐 is the 𝑦-intercept or where it cuts the 𝑦-axis — cept meaning to cut. Now in this video we’re just going to be able to plot graphs and get them into the form 𝑦 equals 𝑚𝑥 plus 𝑐 to be able to plot them. So if we want to plot the graph 𝑦 equals three 𝑥 minus one or any graph for that matter, the first thing we need is a table of values. Well you have to choose at least three 𝑥-values. So we are choosing at least three 𝑥-coordinates. And the reason you choose three is basically just in case we get something wrong. So if you choose two and they give you those two points, we could say okay I’ll draw a line in between them. And here we are, here’s my graph. But actually what happens if we needed the third one. It was here and that tells you either the first, the second, or the third point is wrong. So by doing at least three, it gives us a way of checking what points actually correct into our calculations. Now I like to choose negative one, zero, and one because it involves the least calculation, but you can choose whichever points you like.
So if we do choose negative one, zero, and one, what we need to do is substitute each of those 𝑥-coordinates into our function to find the 𝑦. So in the first case, we’re going to do 𝑦 equals three multiplied by negative one minus one. Well, three multiplied by negative one is negative three. And then subtracting one, we get negative four. So our first 𝑦-coordinate is negative four. And substituting our next 𝑥-coordinate of zero, we’ve got three multiplied by zero, which we know is zero. And subtracting one, so zero minus one is minus one. So our next 𝑦-coordinate is negative one. And then finally three multiplied by one minus one, so three multiplied by one is three and minus one is equal to two.
So now we need a set of axes to be able to plot our points. Now we can see that our table of value leads us to coordinates to be able to plot. For example, the first one, we’ve got negative one as the 𝑥-coordinate and then negative four as the 𝑦. And the next one is zero as the 𝑥 and negative one as the 𝑦. And then finally, we’ve got one in the 𝑥-coordinate and two in the 𝑦. Now plotting each of these, negative one, negative four, then zero, negative one, and then one, two. And then joining them up, we should get something like this that I prepared earlier.
Now in our next example, we haven’t got it in the form 𝑦 equals 𝑚𝑥 plus 𝑐. So we’re going to have to do that first. Plot the graph four 𝑥 plus two 𝑦 equals ten. So as we can see that as I just said it’s not in the form 𝑦 equals 𝑚𝑥 plus 𝑐, and it must do- first do a table of values. So first we want it to be in that form. So basically what we need to do is rearrange this to get 𝑦 as the subject. So what we’re gonna do first is subtract four 𝑥 from both sides. And that will give us just two 𝑦 on the left and then ten minus four 𝑥 on the right. And then this is two multiplied by 𝑦 or the opposite of multiplied or times by is divide by, so you must now divide both sides by two. Be careful as we need to divide every single term by two. On the left-hand side, we will just have 𝑦 now. Now we’ve got five minus two 𝑥.
Now although this isn’t in the form exactly “𝑦 equals 𝑚𝑥 plus 𝑐,” we’ve got the 𝑦 by itself and we’ve got the 𝑚𝑥 and the 𝑐 on the same side. So as long as it’s roughly in the same form like that, then you’re okay. So now we’ve got this; we can do a table of values. And again I’m gonna choose the same three coordinates as we did last time, so negative one, zero, and one. And substituting them each individually into our function, five minus two multiplied by negative one. So the two negatives will cancel out giving us a positive two. So the five plus two, and we know the answer to five plus two is seven.
And then for the next case where 𝑥 is equal to zero, so five minus two times zero, well two times zero is zero. So it’s five minus zero, which we all know is just five. And then for the last 𝑥-coordinate of one, we’ll substitute that into the function. So we’ve got five minus two times one. Two times one is two; so it’s five minus two and five minus two is three. So there we have our table of values and we know that the table of values leads us to coordinates: the first one being negative one, seven, then zero, five, and one, three.
Finally, we just need to plot the points and join them up with a line. So looking for negative one, seven will be minus [minus one] in the 𝑥 and seven in the 𝑦. Then zero, five, zero in the 𝑥, five in the 𝑦, and then one, three, so across one and up three. And then joining them up, it gives us our straight line here. So we can see it goes through five on the 𝑦-axis and it goes down. And this down is related to what’s in front of the two, being a negative. So this goes on to gradients. So there we have it. To plot linear graphs, what we need is a table of values. But before it’s in a table of values, we must rearrange it to the form 𝑦 equals 𝑚𝑥 plus 𝑐. Now once we have our table of values, we have some coordinates. Obviously we’ve chosen the 𝑥-coordinates, but the 𝑦-coordinates we’ve got from substituting the 𝑥-coordinates into the functions individually. And then these give us the coordinates to plot the points on the graph. And once we’ve plotted the points, we get a straight edge or a ruler which I haven’t done here and draw a nice line through all these points, giving us the function.