Lesson video

Welcome to today's lesson.

My name's Ms. Davies, I just want to start by saying well done for choosing to learn using this video.

I hope you find lots of interesting bits as we explore this lesson.

Let's get started.

Welcome to today's lesson on the intercept point.

By the end of the lesson, you'll be able to calculate the intercept point from a graph and from two coordinate pairs.

There's a few key words that we're gonna be using today that you should have come across before.

Just pause the video, read through them, and make sure you're confident before we move on.

So this lesson's gonna be split into three sections.

We're gonna start by identifying the Y-intercept from graphs and from table of values.

Have a look at the graphs below.

What is the same and what is different? Pause the video and think about your answer.

You might have said something like, all the graphs are straight lines.

They all have the same positive gradient.

They're all parallel.

Some of the differences you might have thought about, they have different heights on the graph.

They are translations of one another.

They cross the axes at different points.

Although they're all parallel, they're all going in exactly the same direction, they're at different points of the graph, and there's different ways to explain that.

We call the coordinate where the graph crosses the Y axis, the Y-intercept.

When any linear relationship is plotted, it forms a straight line.

Almost all straight lines will have a coordinate where they cross the Y axis.

So most linear relationships when you plot it, will cross the Y axis somewhere, even if it's not visible on your limited graph.

If you scroll, it will cross the Y axis somewhere, with the exception of vertical lines.

We're not gonna be talking about vertical lines today, 'cause they don't have a Y-intercept.

Where are the Y-intercepts on these graphs? What I'd like you to think about is what are their coordinates going to be? Pause the video and come up with your answer.

Okay, I've written the coordinates on for you.

So those are the Y-intercepts of each of those graphs.

What do you notice about those coordinates? Yeah, you should have spotted them that the first coordinate, the X coordinate is zero.

And that is always going to be the case for Y-intercepts.

If you think about anywhere on the Y axis, no matter how far up or how far down or even in the middle, the X coordinate is always zero.

(0,0), (0,1), (0,2), (0,5), zero, million, they're all gonna be on that Y axis.

Another name for the Y axis is X equals zero.

And that's because all those coordinates follow the rule the X coordinate is zero.

So if you ever hear the line X equals zero, that is the same as the Y axis.

Quick check then.

Which of these could represent a Y-intercept? What do you think? Well done, (0,0) or (0,5) 'cause they have an X coordinate of zero.

So here's a table of values for the linear relationship, Y equals X plus two.

You may have used tables of values before to plot linear relationships.

Alex says, "The Y-intercept is negative two, 'cause that gives an answer of zero." See where he is looking in his table.

What do you think with his statement? Good spot if you realise he's incorrect.

The Y-intercept has to have an X coordinate of zero.

So in a table of values, we're looking for where the X is zero.

Let's find that.

And then it's a good practise to write Y-intercepts as coordinates.

Yeah, he's right now.

So the Y-intercept is (0,2).

We found where X is zero in our table, and that gives us our Y-intercept.

So the Y-intercept can be found in a table of values where the X value is zero.

I would like you to have a look at these four and see if you can pick out what the Y-intercept would be.

Let's have a look.

So we've got (0,5), (0,-8), (0,0) or the origin, (0,0.

4).

Right, you've got the hang of this.

Now we're looking at where the X coordinate is zero, and that gives us our Y-intercept.

Where would the Y-intercept be if we plotted the relationship Y equals X plus four? If we don't have the graph or the table of values, we can just look at the relationship.

'Cause remember, we said it was when X is zero.

So using our substitution skills, we can substitute zero in for X.

We can rewrite our equation as Y equals zero plus four.

So Y equals four.

So when X is zero, Y is four.

So our coordinate (0,4) would be on that line.

And furthermore that is our Y-intercept.

What about if we plotted the relationship X plus Y equals 10? Again, the Y-intercept is when X is zero.

So let's put zero in for X.

Zero plus Y equals 10.

So Y has to be 10 doesn't it? To make that true.

Zero plus 10 is 10.

Therefore (0,10) follows that relationship.

(0,10) is on that straight line and it is our Y-intercept.

Almost all linear relationships will cross the Y axis when plotted.

We said not vertical lines, but other than that they will cross the Y axis.

This method will work for graphs of most linear equations.

However, some might require more steps.

So let's look at 2X minus Y equals negative three.

What would the Y-intercept be there? Same idea, X has to be zero.

So two lots of zero, subtract Y equals negative three.

Well, two lots of zero is zero.

So zero subtract Y is negative three.

Can you think of a Y coordinate that's gonna work? Yeah, Y has to be three.

So (0,3) is on that line and it is our Y incept.

Okay, what value of Y will make each of these equations true? Pause the video and have a think.

So the top one is gonna be four.

Four add zero is four.

Second one negative four.

Zero subtract four is negative four.

C is gonna be six.

Zero plus six subtract two is four.

And the bottom one zero.

Two lots of zero is zero plus zero is still zero.

Fantastic time to have a practise.

So for each question I would like you to circle the Y-intercept, both on the graph and in the table.

Tell me where to find it.

When you've done that, can you write the Y-intercept as a coordinate? Question three, I'd like you to match each Y-intercept, so that's A, B, and C with a graph, a table, and an equation.

Give that one a go.

Well done, let's have a look at where these Y-intercepts are.

So you should have (0,3), and check where I've circled it in my table and on my graph.

For B, you've got (0,-4).

Check that you've circled the correct part on the graph and in the table.

And C, (0,2.

5).

For three then.

So A, (0,6) should match up with the graph E, the table G, and the equation 2Y plus 3X minus 12 equals zero.

B, (0,12) matches up with graph D, table H, and equation J, Y equals x plus 12.

And C, (0,-12) matched with graph F, table I, and equation K, Y plus 12 equals X.

In the second part of the lesson we can look at calculating the Y-intercept from a table of values, which maybe not quite so obvious.

So Sam is looking at this table of values here.

When I plot this graph, it will not have a Y-intercept.

What do you think to Sam's statement? What you might have noticed is that Sam is correct, that none of the points in the table are the Y-intercept.

However, if Sam plots these points and draws the line of the relationship, there will be a Y-intercept somewhere, it's just not represented in the table.

How could Sam then find the Y-intercept? What do you think? So what Sam could do is plot the points accurately on a graph, draw a line, and then find the Y-intercept Or they could use the gradient from the table.

If we look at the table, we could find the gradient, and we could use that to work out the Y-intercept.

Let's look at this first method.

So we've plotted the points from the table, draw the line, and we can find the Y-intercept.

Now that is a really good method to use if we have an accurate set of axes that we can use, and if the points are easy to plot and easy to join up with a straight line.

However, drawing the axes and plotting the points can sometimes be time consuming.

Sometimes we don't have the resources available, we don't have the graph paper.

And sometimes if our Y-intercept is really large or really small, when we plot the points and draw the line in, the line doesn't hit the Y axis on the scale that we are using.

So it'll be off the graph, and therefore we can't work out what the Y-intercept is.

Let's look at an alternative method.

So this is our table of values on the right-hand side, and I've just added in a couple of columns on the left.

When X increases by one, Y increases by two.

So we can see from the table the gradient of the line must be two.

We can then count back in twos until we know what the Y value would be when X is zero.

The Y-intercept then is (0,-9).

We can do the same for negative gradients.

So here we've got a negative gradient, 'cause as X increases, Y decreases.

Our gradient is negative five.

So if we're working in the opposite direction, we're gonna need to add five, until we get to our Y-intercept, which is (0,19).

Just check that our Y values are all decreasing in five moving from left to right, yes they are.

So the Y-intercept is (0,19).

Have a look at these four relationships.

What are the Y-intercepts for each one? Off you go.

So the first one, we've got a gradient of three, so we need to subtract three to get (0,2).

Second one, we've got a gradient of negative four.

So we're gonna need to add four to get (1,7), and add four again to get (0,11).

So our Y-intercept is (0,11).

C, X is increasing by one, Y is increasing by three.

So to get to zero we need to add on three to negative one, which gives us two.

Our Y-intercept is (0,2).

And for the last one we've got a gradient of negative eight.

So we're going to need to add negative eight, add negative eight and add negative eight again to get from our coordinate (-3,4) to our Y-intercept, which is (0,-20).

So we can work out the Y-intercept if we know any two consecutive coordinates by counting back to an X value of zero.

So Sam's looking at this one.

(10,8), (11,11), (12,14).

Their X values are increasing by one from 10, 11 to 12, and their Y values are increasing by three, eight, 11, 14.

Sam is asking, "Counting back from 10 will take a while.

Is there a quicker way?" Let's have a look.

How many steps of one are there between zero and 10? Well, hopefully you can see there are 10 of them.

So there are 10 additions of one between zero and 10.

So if our gradient was three, there must be 10 additions of three between the Y-intercept and eight.

We don't want to add three on that many times, so we can see that overall this must be an addition of 30, 10 lots of three.

So if we want to work backwards, we need to subtract 30 from eight, which gives us negative 22.

So the Y-intercept would be (0,-22).

We're gonna have a look at some other examples.

So what is the Y-intercept for this table of values? Well, we've got a gradient of five, but we need to know what the Y value is when the X value is zero.

So we need to get up from negative eight to zero.

We can do that by adding on eight times.

So we need to add our gradient to five eight times.

So five eight to 40.

Negative seven add 40 is 33.

The Y-intercept then would be (0,33).

We're gonna have a look at another couple of examples.

So watch the one that I'm doing on the left-hand side, and then you are gonna give it a go on the right-hand side.

So what is the Y-intercept for this table of values? Well, first I need to work out my gradient.

As X increases by one, Y decreases by two.

The gradient then is negative two.

I need to get from eight back to zero to find my Y-intercept.

So I need to do eight lots of two, which is 16.

And then because it is negative two moving right, I'm gonna need to add on 16 if I want to work back to the left.

Five add 16 is 21, so the Y-intercept is (0,21).

If I keep taking away two from 21, I will get to five and three then one as says in our table.

Have a go at this one on the right-hand side, what is the wine set for this table of values? Off you go and then we'll look at it together.

So our gradient is negative four, when X increases by one, Y decreases by four.

To get from 12 back to zero, we need to add on four 12 times.

So four 12 is 48, and we need to add on 48.

That gives us a Y-intercept of (0,50).

Well done if you've got that one right.

Time for a practise.

So for each table of values, have a go at working out the Y-intercept.

For most of them I've drawn a table to help you, and you are looking for when X equals zero.

Well done, let's have a look at our answers.

So for A, you've got a gradient of three.

Two lots of three is six.

So we need to take away six from 13, that gets us (0,7).

B, we've got a gradient of two.

To get from six to zero, we need to take away to six times, let's takeaway 12.

19 subtract 12 is seven.

We've got zero seven for that, second one as well.

C, we've got a gradient of four.

To get from 25 to zero, we need to take off four 25 times.

That's subtracting a hundred.

So we get negative 83.

D, we've got a gradient of negative five from 10 to zero.

That's adding 10 lots of five which is 50.

Negative three add 50 is 47.

Getting a bit trickier there, so I don't know if you've got that one.

E, we're working the other direction, 'cause we've got negative X values.

We have a gradient of negative three.

To get from negative three to zero, we need to add negative three another three times.

It's the same as adding negative nine.

So we get a Y-intercept of (0,0).

That one you might wanna just check by subtracting three from 15, then from 12, then from nine, and checking we get back to (0,0).

F, so we've got (-10,-9) as a coordinate.

In order to get to zero, we need to add our gradient on 10 times.

Our gradient is positive six.

10 lots of six is 60, negative nine plus 60 is 51.

(0,51) for that last one.

Well done, you're really stretching your brains at the end, I'm very impressed.

Our final section of the lesson then, we're gonna look at calculating the Y-intercept just from two coordinates.

So bringing together all the things we've talked about so far.

So we're gonna start looking at some questions where we might have quite a lot of steps to do.

I strongly recommend that you write things down really neatly to show you are working out so that you can follow along with the steps that we are doing.

So we can work out the Y-intercept if we know two coordinates on the line.

So what would the Y-intercept be for the line that goes through (4,10) and (7,16)? I recommend tossing them roughly first so we have an idea of where they are.

But remember, this is just a guide.

We don't know where the Y-intercept would be, 'cause that's the point of the question.

So I don't know whether that line should have a positive Y-intercept a negative Y-intercept, go through (0,0).

So I can draw a line, but I've decided to dot it, because I don't know exactly where that line's gonna go.

I do know that I'm gonna have a positive gradient.

So let's calculate our gradient.

Change in X from four to seven is plus three.

From 10 to 16 is plus six.

So our gradient's going to be two.

Now I'm gonna put my coordinates in a table of values to help me count back.

So there's (4,10), that was one of my coordinates, and I need to work back to zero on the X axis.

So if I know my gradient was two, I need to do four lots of two is eight.

And I need to take off eight to work back to my zero X coordinate.

10 take away eight is two.

So my Y-intercept is (0,2).

What's about the Y-intercept for the line which goes through (11,21) and (15,5)? Let's do the same thing again.

So roughly sketch my points.

I'm expecting the gradient to be negative, although I'm not sure exactly where my Y-intercept's gonna be, but I know that the gradients negative.

Let's look at my change in X.

So from 11 to 15 is plus four.

From 21 to five is minus 16.

To get a change in X of one, I need to divide by four.

So my change in Y is negative four.

So my gradient's negative four.

I was expecting it to be negative, so that's good.

Let's put our coordinates in a table then.

So I'm gonna look at the (11,21) coordinate.

To get from 11 to zero, that is 11 additions of four.

So 11 times four is a 44.

Add on 44, that gets us to 65.

Our Y-intercept then is (0,65) So Laura wants to work out the wine set for the line going through the coordinates, (-10,41) and (-8,31).

Here's her working out.

She's plotted a couple of coordinates.

Do you think they're correct? Then she's looked at her change in X, is that one correct? Her change in Y? And she's put it into a table of values.

You might wanna pause and see if you're happy with what she's done so far or not.

Then she goes on to divide.

And state that the gradient is negative five.

Putting it in a table.

She works from negative eight up to zero.

And says that the Y-intercept is (0,-9).

If you haven't already, just pause the video.

Do you agree with Laura? Is she correct? Before you gimme your answer, Alex weighed in.

He said, "You must have made a mistake because the graph says the wine is positive." Remember that graph we drew at the beginning? What is your final answer? Who do you agree with? In this case Alex is incorrect.

The graph was just a rough sketch.

So be aware that if you draw a graph, we don't necessarily know all the key features of the graph if you've just sketched it roughly.

It did help us work out the gradient, but we don't know that the Y-intercept's gonna be positive.

In fact, Laura's working out on the previous page was spot on.

We can check our Y-intercept works by seeing if it works for the second coordinate too.

What is the Y-intercept for the line which passes through the coordinates (8,1) and (12,3)? So we have a gradient of a half.

If we use the coordinate eight, eight lots of a half is four.

So we need to subtract four.

That gets us a Y coordinate of negative three.

So our Y-intercept, (0,-3).

Now we can check using the other coordinate.

So we said the gradient was a half.

12 lots of a half is six.

Three subtract six, negative three.

So Y-intercept is (0,-3), which is the same answer we had before.

So it looks like the Y-intercept is (0,-3) as it works with both coordinates and the gradient of a half.

Now this is a really useful way of checking some of your working.

We know in maths that having ways of checking things can be really helpful.

However, it is possible to get the same incorrect coordinate for the Y-intercept, if you've made multiple mistakes.

So just be aware there is that limitation.

What I recommend you doing is check your working for the gradient before you're going any further.

So double check that your gradient is correct before looking at your coordinates.

So Laura has tried to work out the Y-intercept for the line passing through at (7,15) and (10,21).

See if you can identify any mistakes.

She's got an answer of zero negative nine for both of her coordinates.

Can you spot the mistakes? Well done.

Notice that because Laura has set her working out really clearly, it's easy to spot where things have gone wrong.

So to start with, between seven and 10, that should be plus three, not plus two.

What that means is when we divide them both by three, our gradient is two.

So Laura has made a mistake with the gradient to start with.

Then here, three times seven isn't 24.

So she's made a mistake with her multiplication.

But that's not actually supposed to be a three is it? 'Cause we said that the gradient's actually two.

So it should have been two multiplied by seven, which is 14.

So that's two mistakes there.

Those two mistakes have then meant that she's got the same answer twice, but it's the wrong answer.

Well done if you finished her working out to show that the answer should be (0,1).

Right, well done for that lesson so far today.

That's lots of concepts that we are bringing together.

For your practise, I'd like you to have a go at working out the Y-intercept.

I've given you two coordinates, and I've given you a bit of a sketch.

Work out the gradient, then work out the Y-intercept.

Off you go.

Well done, for this second set, I haven't drawn the graphs for you.

There is space for you to sketch your own if you wish.

Off you go.

Well done, that was getting tricky towards the end.

Hopefully if you presented your working outs in the same way as Laura has been doing, then you were able to follow those steps.

So first one, we get a gradient of three.

Five threes are 15.

So our Y intercept is (0,5).

B, we have a gradient of five.

Five twelves is 60.

52 subtract 60 is negative eight.

C, we have a gradient of negative two.

Four twos, eight.

We need to add on eight to negative 10 to get negative two.

There's quite a few elements there that it's easy to make mistakes on.

So if you didn't get the right answer, just follow those workings through again, see if you can spot where you've gone wrong.

This second set.

So from 10 to 18 we've increased by eight.

But from eight to zero we've decreased by eight.

So our gradient is negative one.

Our Y-intercept then.

10 lots of one is 10.

We need to add 10 onto eight, so we get (0,18).

For this middle one, seven to 10 is an increase of three.

54 to 75 is an increase of 21.

Our gradient then is seven.

Seven seven's a 49.

So we need to subtract 49 from 54, and we get a Y-intercept of (0,5).

Last one, from six to nine, we've got an increase of three, and 13 to 22, we've got an increase of nine.

Our gradient then is three, six threes at 18, 13 subtract 18 is negative five.

So we have a Y-intercept of (0,-5).

Incredible work there guys.

I know there were some trickier bits, okay? So I'm really, really proud with how you've kept going with that.

Hopefully there's enough information there, and that if you have made some mistakes, you are able to go back through it and see where you've gone wrong.

Of course, you can always use a calculator to check any of your mental arithmetic as well.

So things we have learned.

The Y-intercept is the coordinate where a graph crosses the Y axis.

The Y-intercept can be found from a graph, a table, or just from a relationship.

The Y-intercept can be found from two coordinates by finding the gradient between them.

And all those skills that we used today in that final section of the lesson, were built up from things that you were already knew, and will build towards even more skills if you carry on with any of your graphing skills.