In this AQA GCSE Maths video, we take a look at straight line graphs and how to find their y-intercept and gradient. We also look at how to determine whether two lines are parallel or not.
Graphs are used to show the relationship between two different variables and are found throughout the world in everyday life, whether they are seen in newspapers, businesses or the media to help indicate data in an easy-to-understand way.
The most common professions that usually use graphs involve maths and science.
So we're looking at straight line graphs, and we're first going to understand the standard format. So this slightly weird looking Y equals MX plus C equation, we're going to look at the gradient and the Y intercept and what to actually need. And finally, we're going to do a couple of examples of parallel and perpendicular lines, once we kind of understand what the gradient is telling us. And we're going to finish off with an exam style question. So let's first look at this standard equation.
Y equals ms plus C. What does it mean? Why do we use it? Well, the easiest way to understand this is if we look at an example. And so if I draw some random line, I'll do that, we're going to have a yintercept.
So we're going to intercept the Y axis. At some point, we can say this is equal to one. And so in this case, c would be one, right? Essentially, where do we cross the Y axis? What's the value of Y when we cross the axis?
The gradient is a bit trickier. If we were to actually look at what's going on in this equation. Imagine if I increase x by one.
Say I start at this point here, right? I start at my intercept and I increase x by one. So I move along my x axis. This is my x and my Y axis. If I'm a long one in the x direction, let's say m is equal to t.
What's going to happen to Y if I increase x by one? What? If I increase x by one, this whole thing is going to increase by t because I'm timing this increase of one by t because m is two in this case. And so what we end up seeing is on my graph, my Y is going to increase by two. That's why I move up by two spaces.
And again, I can do the same thing again, right? I can move along one in the x axis. And again, my equation tells me I have to move up in the Y direction by t. And that's what m is, right? M is really telling me how much do I move in my Y direction, given that I move some distance in the x direction.
And this is easier to understand if we decide that the change in x, this is change in Y over change in x. If we decide the change in x is just going to be one, where in this case we just said the change in x is one in both these cases. Well, all of a sudden I basically end up with m is equal to the change in Y.
If we decide that we make the change in x equal to one, right? Because anything is going on, one is just itself. So that's my gradient, right? It really tells me how much do I move up or down by, and therefore how steep is my graph. If we imagine instead of a gradient of t, we had a gradient of four, well, what's going to happen is, rather than moving up by two, I'll move up by four.
I'll move up to that point there and we end up with a line that's not supposed to be wiggly, but it's slightly wiggly, went up with a line that's much steeper than my original line. So that is gradient y intercept. Again, it's just where we intercept the y axis. And so that's been taken care of. Let's now look at parallel and perpendicular lines.
Well, parallel lines are very, very easy. All we need to understand is if I keep this same example that I had over here, this is going to have just the finishes off. We decided that two and the intercept is one. So the equation for this would be y equals two x, because m is two plus one, because c is one. And so I'm just going to copy this out over here.
This is my equation for the line two x plus one. And let's look at a parallel line. So a line that's going to run parallel to my original line. Well, because it's parallel, it's going to have the exact same gradient and same slope, right? Because if they're pointing in the same direction, they've got to have the exact same slope.
And because gradient is literally the definition of the slope of the line, the gradient has to be the same. And so for this line, my gradient is t, right? I'm going to say m is t. We often use m for gradient. And for this line, again, I know my gradient has to equal the exact same thing.
It has to be equal to slightly weird, too. There we go. Perpendicular lines are slightly trickier, but again, not too bad. And we'll stick with the same example. We might as well stick with the same example.
I'm actually just going to keep the gradients for now, so I'm going to say just the gradient of this line and let's look at a perpendicular line. So I'm going to draw this in a new colour. I could have a perpendicular line anywhere along this line. All that perpendicular means is we intercept our right angle. So that angle there between the two lines is 90 degrees.
Well, what's the gradient of this intercept of this perpendicular line going to be? Well, I know that because my original line is pointing upwards, my new line has to point downwards. We can't have a 90 degree angle and have two lines both pointing upwards. It's just never going to happen. So I know that my new gradient is going to be minus and it's going to be minus some number to understand what that number is, to understand what the magnitude of this gradient of this line is, it's probably easiest if we do another example.
And so I'm going to draw in a shallower line in this case, I'm going to draw in a shallow line. Let's make it look like this. And again we'll plot out the perpendicular line. So again, it doesn't really matter where on the line I draw my perpendicular line as long as it's perpendicular. But what we're seeing is this new line could have, say, a gradient of what we work at, right?
We could say if we go along by one in the x axis, how much do we go up by in the Y axis? Let's say we go up by half. That's about right. So my gradient is going to be a half. What I'm doing is from my original line to my new line, I'm reducing my gradient, I'm making it shallower.
What was happening to the perpendicular lines? Well, originally, the perpendicular line was quite shallow. This is my original perpendicular line. And what we can see is as we go from my first line to my second line, my perpendicular line is actually increasing in gradient. It's getting steeper, isn't it?
And so anything that's steeper has a much greater gradient. So what's happening is as I'm reducing my original gradient, I'm increasing my perpendicular gradient. Well, the way that we represent that mathematically is we do one over the original gradient. And I'm actually going to label the new gradient of perpendicular line MP. So we can get confused.
These are two separate M. This is my original, I can call this Mo and this is MP. What we find is our new gradient. Perpendicular line is always going to be equal to the negative reciprocal. So minus one over the original gradient.
And we can see in practise, if I reduce the value of my original gradient, if this goes down, what's going to happen to the overall fraction? If the denominator decreases, the overall fraction is going to increase. And that is current on topic designs. So let's dive into a couple of example questions. Now, hopefully this all makes sense.
So we've got an example question. It says on the grid, draw a graph of Y equals three minus two x, from minus two to four. So this isn't in our standard equation format, right? We generally want Y equals MX plus C. So what we can do is we can just swap these around.
I can put my minus two X first and my plus three second, and all of a sudden we can see exactly what's going on here. So I know my gradient is going to be minus two and my Y intercept is going to be three. And so I know my line has to pass through that point there on the graph. I have to pass three Y equals three. And the gradient tells me that if I go along one in the X direction, I've got to go down by two in the Y direction.
So I get to that point there. Again, I go along one and I go down by two. And I can keep doing this, but I don't need to draw all the points. I can actually just jump straight in and draw a nice straight line and I get that and we can see it intersects all my points. So we really just need two points.
I could have actually skipped this point or skip this point, it doesn't really matter. That is online. So as soon as we understand the gradient and the wintercept, we can then draw online. Let's do one or two more. A straight line, l is shown on the grid.
Work out the equation of INL again. I know jump to more contrasting colour. I know that the Y intercept is going to be one, so C is going to be one in this case. And if I look at what's happening, if I go along one in the x direction, how much do I go up or down by in the Y direction? Well, I know in this case I've got to go up by T in the Y direction, and so that tells me my gradient.
And so if M is t and C is one, I end up with Y equals MX plus C. So M is going to be t and C is going to be one. Right, last example. And then we'll call it a day. So the straight line l has equation Y equals to minus five.
Find an equation for the straight line perpendicular to l, which passes three minus t three. So in this case, I'm just going to draw out what's going on. So I think it's easier if we can visualise it. And two x minus five. But I'm going to intercept at minus five and I'm going to have a gradient of two x.
So it's going to look something like that. That should really intercept that. And I want to work out the perpendicular line that passes three minus two three. But if it's passing through minus two three, it's going to pass through that point there, roughly there, and it's going to be perpendicular. Can't draw it.
So it's going to look something like that. That's going to be my perpendicular intercept. Well, I know that the gradient of my new line, a perpendicular line, is going to be equal to the negative reciprocal of the original gradient minus one over the original gradient. Well, the original gradient, which I call Mo, is going to be t, so it's minus one over t and we're halfway there. Again, we always need our line equation to be in this format, y equals MX plus E.
And so I'm halfway there. I've got M, which is minus a half, but I don't have C. I need to find out what is C. How can I do that? Well, I've got a point on the line, right?
I know the fact that my permanent line goes through minus t three, and it goes through this point here, minus t three. And so I can use that to work out what T is. Because I know that if I put in X equals minus T, these are my x and Y coordinates. If I put an X equals minus T into my equation, the equation should always split out three at the end of the day because I know the Y has to equal three, and so I know that I should be able to get plus C to get three. Well, minus one two times minus two, the minus cancel out.
We've got a half times two that could be one. So one plus C has to be three. Well, C therefore has to equal two.
And so we end up with Y equals minus one, two x plus T. And that looks about right, doesn't it? That point there is going be roughly two, and that is the end of.
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