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GCSE Maths graphical problems questions will commonly be linked to real-life scenarios and situations such as travel graphs, temperature graphs and conversion graphs to name a few. This video breaks down the common framework used to tackle graphical problems.

The scale of graphs is important so it is essential to have a basic understanding of the scale and what each notch on the axes equals in order to solve the problem correctly.

Graphical problems are very common in math in particular, also appear a lot in the sciences. A graph is just a good way of representing two different pieces of information against one another. And the skills you learn here are ones that you might take all the way at university if you pursue something like a scientific career. In general, graphical problems, I've split into these different sort of topics, but they're all strongly related to one another. The first one here is graph equations, okay? Because we could always have a graph, which is just two pieces of information here, expressed usually by what we call the Y axis and the x axis. And in both of these cases, we can have some kind of way of relating these two bits of data. Now, the X could be anything we want to really. A math, we just keep it as x. But in, say, for example, physics, it could be something like, say, distance. And then if we had this Y one here, we could call that maybe say, velocity and get a slightly weird looking time graph. And we're always going to have some kind of relationship between Y and x.

And the most common graph equation is that Y is going to be equal to x, which is then multiplied by some value, which we call the gradient. This is represented by the letter M, okay? And then that will give you an idea of how much Y increases compared to x. But of course this means if we substitute in x equals zero, which Y should then also equal zero. It's clearly not sure in this graph because when x is equal to zero, which corresponds to here, y is equal to some nonzero value, we call this value C, which is the Y intercept.

Okay?

So the full equation here is y equals MX plus C. Okay? So the way you figure out C is you figure out what is x when it is zero. And the gradient you have to calculate as well. If we actually look at this equation here in a weird way, we could think about dividing by x. We could say that Y divided by x is going to equal well, it's going to equal M plus, and then we get the C over x. We're not too worried about that because the C is just a constant that gets encoded in there, so it's worried about it. So the way we calculate the gradient is a change in Y over change in x, because that change is what gets rid of that C term. And the way you do that is literally measure off the graph. You figure out what is the Y value here we'll call that, say, Y two. What is the Y value here we'll call it Y one. And then similarly, when you've got these two different values here, we've got a corresponding x value and Y values. This would be x one and x two.

Okay, divide one by the other and we'll get the graph. So we'll do that with an actual practise in a minute for finding the tangent very common skill. The next one we're interested in is tangent, a word I just mentioned. So tangent just means a straight line coming off of something. And when you have a graph that is not necessarily a straight line, we're just going to go for a graph here where we've instead got this sort of nice looking curve. So immediately obvious how you figure out what the gradient is. Because of the gradient, I just said change in Y over change in X. However, it's curving during that section, so it's not like we can put one point to another easily. So instead of tangent is where you draw a straight line coming off of this particular shape here, and that straight line should match up here with one of the points on the graph. That straight line comes off of that graph itself. So it's at this point here and then it comes off of that graph and they form a 90 degree angle. This is one of those things you get very good at with practise, but generally take the ruler and just draw a straight line.

So if you zoomed in really hard, it was just going to touch that point. A tangent should also generally not touch the rest of the graph if you've got something like a square graph, but it inevitably will if you've got a more complicated graph. Again, something we'll get with some more proper practise when we do the exam question. The last one here we have is rates of proportion, which is a way of rather than using graphs, but instead we're going to be talking directly about the equations that link to pieces of data. For example, on that last one, that was a square graph. We might say Y equals x squared and we might then consider what this graph looks like in comparison to say, the opposite, y equals the square root of x, because different systems in the world have different rates of proportion and we know how we need to know how to represent all of them. So we're going to move straight on to exam questions here because the exam questions are pretty much the basis all these examples anyway. So this is not an exact exam question. This is one that I pulled off of BBC byte size.

But it's a very close approximation for the kind of questions that you'll get for straight line graphs. You might be asked, what is the equation? This graph just a reminder, the thing that we're heading towards is y equals MX plus C is the yintercept and m is the gradient, m is how much is y changing when we change x.

Okay?

Now uniquely, this graph is actually going down as X increases. As X increases this way Y is going down. So we know this is going to be what we call a negative gradient. Basically, it's one that says if we increase X, y has to go down. So M must be a negative number. We'll find that in a minute. So to find C, we have to figure out what is Y when x equals zero. So if we go to x equals zero here, we can see the Y is equal to two.

Okay?

So when X equals zero, y equals two. So C is equal to two.

Okay?

Now to find the gradient that M, we have to do the change in Y divided by the change in X. Now, in the case of a straight line like this in math, which has been made on some graph paper, we can afford to be a little bit less precise. However, the reason you do this in a scientific scenario is you'd want to pick a very wide range of data. This minimises the kind of errors you're going to make and it's something that we should do here just as good practise. So I'm going to draw a triangle here that clearly links up the entire graph. So we're going to do the change in Y.

Okay?

So Y over here is plus five. So Y two up here is equal to plus five. Y one down here is equal to minus four. The change in Y is going to be how much? There is a difference between the two of these values, which is going to be nine. If you wanted to show that using an equation, you do five minus minus four because the difference is the two of them subtracted. But this minus four here contributes a minus five. So change in Y is equal to nine. We then do the similar process for the change in X. At this point, I'd recommend maybe pausing the video and see if you can do this by yourself if you're unsure. But change in X is going to be equal to so we've got X.

One over here and then we go extreme over here.

I've labelled this the wrong way around what you might see in the textbook. It doesn't matter. As long as you're consistent, it doesn't matter. So in this case we have four, and in this case over here we have minus two. So the change in X is going to be well, we know there's six squares between them, so it's going to be six. But again, if you wanted to represent that, you do four minus minus two.

Okay?

So the change in Y for the change in X is then going to be six.

Okay?

So the change in X is six and the change in Y is nine. But I've not been fully correct here because I mentioned before the M has to be a negative number. In this case. This is a fairly classic trap to fall into when you're doing things that are in math. So instead, what we have to do is make sure that either M is a negative number and recognise the shape. If you think about it with shapes, that might be the better way. Alternatively, for really being consistent about it, what we should have done is if we're doing four minus minus two, we should have done minus four minus five for delta Y. So change in Y is actually negative nine, because as X has increased by six, y has decreased by nine. Okay, so our full equation is Y is going to be equal to and then we get minus nine divided by six x plus C. It's common practise to simplify our equation. So we get Y minus three over two, x plus C. And that's the equation of this graph here. That's how you determine it. That was very long winded. I wanted to make sure you had every single step.

I will say that as you get more practise with this in biology, chemistry, physics and math, you'll get very quick at doing these graphs and you'll decide on your own personal method and ways that you would write down these different variables. But you will get very good at this. So, tangents, a container is filled with water in 5 seconds. The graph shows a depth of water, d centimetres at times T seconds.

Okay.

The water flows into the container, a constant rate, which diagram represents the container. So in this form, we have to do a bit of visual imagery, which is very common in graphs because it also relates quite closely to shapes. So, as we can see, we have a graph that initially was going at a constant rate and then it slows down. The way I like to think of this is that between three and 5 seconds, we only really change the depth by five centimetres, which is terrible. It's not really much in comparison to earlier, where between zero and two we change it by 15. So we want a shape where we get an initially large amount of water, then it slows down. So A would be constant, that would be a straight line, d would have a weird wiggly shape, b would end up increasing as time goes on because the depth is easier to fill. So it's actually C in this case, where you have to have more and more water to increase the depth. So the answer for that is C. And use the graph to estimate the rate at which the depth of water is increasing at 3 seconds.

And the way that we do that is at 3 seconds, that corresponds to this point here. And we'd have to draw a tangent and make this into a nice red colour so it's clearly visible. We draw a tangent for 3 seconds, that point there, okay? And that tangent is going to be parallel to the surface at that point, but otherwise makes a 90 degree to it. And then it makes a 90 degrees to the rest of the curve, I should say. So then how we figure out the Rate of change of Something is the gradient. So at that point, we just take a gradient again. So we go from this point down to this point and then if I pick another colour here, these two points here, okay, the change in Y is going to be equal to let's take A look at 25 -15 does equal to ten and change in X, which is equal to four minus one which equals three. So we can estimate that the rate of change in this case is going to be equal to Ten over three, which is about Equal to 3.3. And that Means that The Water is Increasing by 3.3 centimetres every second at this point.

The last one is rates Of Proportion, which is one interesting one, which is which Graph represents The Information where we have Y being Directly proportional to the square root of the distance X. So Y is proportional to the square root of X. That's Another Way of saying it's equal to but multiplied by another number. But when we put into a graph, it just gives a shape. This is A useful trick you guys might not know for the square Root of X, what we can do is draw on What Y equals X squared would look like, which will be something like this, okay? And then to take the inverse of A graph. So the opposite operation, which in this case would be Y, equals the square root of x. We'd actually draw a straight line from the origin and Then we'd see the reflection. Now, I've already drawn this on C, because, as you can see, these two are a reflection of one another. As we end up seeing Y equals x squid increase, this graph here will decrease at the same rate. And so the answer in this case is C. That's how you determine the proportionality of these.

I want to leave this last one we have A Challenge for you guys. The 100 miles journey costs £36. What's the Cost of a 250 Miles journey? So we got to think is that in this case Y 100 Miles is Equal to the square Root of 36 times by some number. We'll call it, say a and then what you got to figure out is what's it going to be for 250 Miles given it's going to be the same a square root of some answer. Anyway, guys, I encourage you to give your answers at the bottom. If you've got any questions, please feel free to also put them down there. I'll answer them. And hopefully this helps you with answering and graph questions in math and subjects beyond.

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