Lesson video

Hello, Mr. Robson here.

Welcome to maths.

Today's lesson is all about equations and their graphs.

Love an equation, love a graph, so let's not hang around.

Let's get stuck in.

Our learning outcome is that we'll be able to recognise the determining features of the shape of a graph from its equation.

Key words that will come across today.

The relationship between two variables is linear.

If when plot on a pair of axes, a straight line is formed.

That's linear.

The gradient is a measure of how steep a line is, the word gradient.

An intercept is a coordinate where a line or curve meets a given axis, intercept.

Watch out for those keywords throughout today's lesson.

Two parts to our lesson today we should begin by reviewing features of linear graphs.

We can plot the graphs of equations.

One method is to populate a table of values.

For example, if we wanted to plot Y equals 2X and Y equals X squared, we could start with those tables of values and we could populate the Y coordinates for those respective X coordinates.

When we do that for Y equals 2X, we get this line.

That's a linear equation.

It's a linear graph.

Y equals 2x.

When we do it for Y equals X squared however, we get these values which generates this line.

This is a quadratic equation.

Y equals X squared.

We call that a quadratic.

It creates a non-linear graph.

We call this particular curve a parabola.

Most importantly, it's nonlinear.

Key features of the equations affect the shape of the graph that is formed.

For example, an X exponent of one gave us a linear graph.

An X exponent of naught, one gave us a non-linear graph.

Quick check you've got that.

Which of these equations will form a linear graph? Is it A, Y equals three X plus two? B, Y equals X cubed plus two or C, Y equals 3x minus two? Pause this video.

See if you can spot the linear graphs.

Welcome back.

Let's see how we did.

I hope you said A, Y equals 3x plus two will form a linear graph.

The X exponent is one.

It's a linear equation.

It will form a linear graph.

B, however, will not, the X exponent is not one, so Y equals X cubed plus two will form a non-linear graph.

C is another linear graph.

The X exponent of one linear equation, linear graph.

How will these two linear graphs look the same and how will they look different? Y equals 3x plus two and y equals 3x minus two.

Let's start by populating the tables of values.

They're the coordinates for Y equals 3x plus two.

They'll give us that straight line.

The coordinates for Y equals 3x minus two, giving us that straight line.

The key question, how do these two linear graphs look the same and how do they look different? Pause this video.

Point out one similarity and one difference.

See you in a moment.

Welcome back.

I hope you said both graphs have the same gradient.

They're parallel lines, the same gradient.

What's different about them? The graphs have different Y intercepts.

Gradient and Y intercept are two key features of a linear graph.

Y equals 3x plus two is a linear graph with a gradient of three and a Y intercept of zero, two.

A gradient of three means a change of three in Y for every change of positive one in X.

We can see that on the graph, everywhere we look.

It's visible not just on the graph but in the table of values and in the equation itself.

That change of positive three for every change of positive one in X, we see it in the table of values.

The Y coordinates generated there and we also see the three as the X coefficient in the equation Y equals 3x plus two.

A wide set zero, two means the graph cuts the Y axis at this coordinate zero, two.

It's visible not just on the graph but in the table of values.

For example, the X input of zero gave us a Y output of two, but also in the equation itself Y equals 3x plus two.

Quick check you've got that now.

I'd like you to identify the gradient of this linear graph.

Is it negative two, two or four? Pause, tell the person next to you or say your answer aloud to me at the screen.

See you in a moment.

Welcome back.

I hope you said it's not negative two.

It is positive two and not negative four.

Why is it positive two? Because we can see on the graph there's a change of positive two in Y for every change of positive one in X.

Another quick check, I'd like you to identify the Y intercept of this same graph.

Is the Y intercept the coordinate zero, four, the coordinate negative two, zero or the coordinate four, zero? Which is it? Pause this video, tell the person next to you or say your answer aloud to me on screen.

Welcome back.

Let's see how we did.

I hope you said it's A, the coordinate zero, four.

It wasn't B, it wasn't C.

It was that moment there.

The line cuts the Y axis at zero, four.

A gradient of two and a Y intercept of zero, four is unique to this line and no other line.

A gradient of two and a Y intercept of zero, four makes this line, the line Y equals 2x plus four.

That's the gradient of two represented there in our equation and that Y intercept zero, four represented there in our equation.

In the form, Y equals MX plus C, it's the value of M, which defines our gradient and the value of C, which defines our Y intercept.

For example, this line we see a gradient of negative four and a Y intercept of zero, seven, so we would call this line, the line Y equals negative 4x plus seven.

You might see that written as Y equals seven minus 4x for efficiency purposes, but the principle still applies.

It's M, the coefficient of X, which determines our gradient and the constant C, which determines a Y intercept.

Quick check you've got that.

This line has a gradient of five and a Y intercept of zero negative, four.

What is the line? Is it y equals negative 4x plus five.

Y equals 5x plus four or y equals 5x minus four.

Pause this video.

Tell the person next to you or maybe tell me at the screen.

Welcome back.

Let's see which one it was.

I hope you said it's not option A.

It's not option B.

It is indeed option C.

The line Y equals 5x minus four.

Why? because it's in the form Y equals MX plus C.

M, in this case, five, defines our gradient and C, in this case, negative four, defines our Y intercept, the coordinate zero negative, four.

The gradient won't always be an integer.

It's important we recognise that.

The gradient is calculated by finding the rate of change in the Y direction with respect to the positive X direction, any two known coordinates enables us to find the gradient.

In this case, we know those coordinates, zero, one and four, four.

We can calculate the gradient by taking the change in Y and dividing it by the change in X.

In this case, a change in Y of positive three, change in X of positive four giving us the gradient, 3/4.

We didn't have to use those two coordinates though, any two coordinates enables us to find the gradient.

We know these two coordinates, zero, one and eight, seven.

Our change in Y this time is six.

Our change in X is eight, six over eight.

I hope you're shouting at the screen now.

That will cancel to three over four.

There we go.

The exact same gradient, 3/4.

With a Y intercept of zero, one.

This must be the line Y equals 3/4x plus one.

Quick check you've got that.

I'd like you to identify the gradient of this linear graph.

Is it negative 2/3, positive 2/3, or negative three over two? Pause.

Tell the person next to you or say your answer aloud to me at the screen.

Welcome back.

How did we do? Did we say it's option A? It's not option B, not option C.

It's the change in Y divided by the change in X.

With respect to the positive X direction.

We're moving positive three in the X direction and negative two in the Y direction.

That's why it's negative two over three.

Practise time now.

Question one, gradients of six and a Y intercept of zero, negative two is a description unique to graph A.

Can you write descriptions for graph B and for graph C? Pause and write those descriptions down.

For question two, there's four lines on that graph.

Can you match those four linear equations to their respective lines? Pause and do that now.

Question three.

Laura is writing the equation of this linear graph and she says, I see a change of three in Y and a change of two in X and an intercept of four.

So this is the line Y equals 3/2x plus four.

It seems like good reasoning but there's something wrong with it.

I'd like you to write at least two sentences to explain Laura's errors to her.

You may choose to annotate the graph as part of your explanation.

Pause, write a couple of sentences and perhaps annotate your graph now.

Welcome back, feedback time.

I gave you a description which is unique to graph A and asks you to write descriptions for graphs B and C.

So for graph B, did we write a gradient of positive two and a Y intercept of the coordinate zero, negative one? For C, did we write gradient of negative three and a Y intercept at the coordinate zero, two? I hope so.

For question two, I asked you to match linear equations to their respective lines.

That one was line A, a Y intercept of zero, three with a gradient of positive two.

There is line B, this same Y intercept of zero, three, but a gradient of negative two.

There's line C, a gradient of negative two again, but with a Y intercept of zero, negative three, meaning this must be line D, the same Y intercept zero, negative three, but a gradient of positive two this time.

For question three, I asked you to write at least two sentences to explain Laura's errors to her and perhaps annotate your graph when doing so.

Say, we might start with explaining the gradient.

The gradient is the change in Y divided by the change in X, but with respect to the positive X direction, the graph currently doesn't tell us which is the positive X direction, so maybe we should change those markings on the graph to look like this.

We can now see a change of positive two in X and negative three in Y.

In the form Y equal MX plus C, the C relates to the Y intercept where we intercept the Y axis, not that of the X axis, so it was that coordinate we were interested in when defining this line.

All that put together tells us that this is the line Y equals six minus 3/2x.

Onto part two of the lesson now, where we're going to be sketching linear graphs.

Having a good understanding of Y equals MX plus C enables us to quickly sketch linear graphs.

So Y equals 4x minus five.

What do we know about it? It's a gradient of positive four and a Y intercept at the coordinate zero, negative five.

Y intercepts in the negatives and it's a steep positive gradient, so the line must look something like that.

When sketching graphs, we label key features.

What key feature do we know? We know that Y intercept, zero, negative five, so we'd label that part.

A sketch is not a hundred percent accurate.

I haven't put a scale on those axes for instance.

It's just an approximate representation that we draw to help us visualise and therefore solve problems. June and Sophia are discussing this linear graph.

June says this looks perfectly diagonal.

That's a gradient of one.

The Y intercept intercept looks about zero, five.

I think this is the line Y equals X plus five.

Sophia says it might be y equals X plus five, but it could be anything.

We have to be careful.

Who do you agree with? Pause this video.

Tell the person next to you or have a good think to yourself.

Who are you going with? June, Sophia? What do you think? Pause now.

See you in a moment.

Welcome back.

I wonder what you thought.

I wonder if you had some arguments of your own to add with regard to what June said.

Maybe you spotted June might be right, but we don't have enough information to be sure.

The only thing we do know, the Y intercept has a positive Y coordinate and the gradient is positive and June had that right.

However, the scale on these axes could be anything, that Y intercept could be zero, five, it could be zero, 500.

We just don't know, we need more information.

Laura said it might be X plus five, but it could be anything, that's not strictly true.

Sophia's right to issue caution over exact accuracy.

However, she's wrong to say it could be anything because we do know the Y intercept has a positive Y coordinate and the gradient is positive.

If we know one line, it can help us to know something about another.

For example, if we knew that this was the line Y equals X plus five, what do we know about this line? Pause this video.

Tell the person next to you or maybe say your answer to me at the screen.

Welcome back.

I hope you said, it's the same Y intercept of zero, five but not the same gradient.

It's a lesser gradient but it's still a positive gradient.

We know the line Y equals X plus five has a gradient of one.

This line is not as steep but it's not a negative gradient, so we do know the gradient of this line must be somewhere between zero and one.

Quick check you've got that now.

Here's a linear graph.

What could the equation of this line be? Could it be Y equals 3x plus four? Y equals 3x minus four? Y equals three minus 4x or Y equals 4x minus three? Which one could it be? Pause.

Have a think.

See you in a moment.

Welcome back.

I hope you said it couldn't be option A.

Our line has a negative Y coordinate for the Y intercept.

It could be option B.

That's a positive gradient with a negative Y coordinate for the Y intercept.

Couldn't be option C 'cause our line's got a positive gradient, not a gradient of negative four, but it could also be option D.

That's a positive gradient and a negative Y coordinate for the Y intercept again.

So two things that it could be.

What could the equation of this line be? Could it be Y equals five minus 3x? Y equals 34 minus 3x? Y equals 17 minus 3x or Y equals negative 3x minus 17? Pause, tell the person next to you or say your answer aloud to me on screen.

Welcome back.

I hope you said it could not be option A.

Our line does not have a Y intercept of zero, positive five.

Hope you said it could not be option B.

Our line does not have a Y intercept of zero, 34.

I hope you said it could be option C.

That's a negative gradient with a Y intercept of zero, 17.

That could be our line, and I hope you said it's not option D.

Our line does not have a Y intercept of zero, negative 17.

All four equations had a gradient of negative three, which this line could be, but only one had the correct Y intercept.

That's why we label our sketches.

Practise time now.

Question one.

I'd like you to sketch the lines Y equals 2x plus 10, Y equals five minus 4x and Y equals 2x minus five on the below axes.

Just sketches.

They don't have to be a hundred percent accurate.

Don't go labelling those axes, however, do label the key feature that is the Y intercept.

Pause.

Give this a go now.

Question two.

Four linear equations, I'd like to match those linear equations to their respective lines on those graphs.

Pause, match them up now.

Feedback time.

I asked you to sketch the lines of Y equals 2x plus 10, Y equals five minus 4x and Y equals 2x minus five.

I hope you included labels for your Y intercepts.

Your Y intercepts should be labelled and zero, 10 should be twice as far from the origin as zero, five and then zero negative five should be the same distance from the origin as zero positive five.

The lines Y equals 2x plus 10 and y equals 2x minus five should have the same gradient.

Therefore, they should look parallel in your sketch.

And Y equals five minus 4x is a steeper gradient, albeit a negative one, but it's steeper than a gradient of two.

This should be visibly obvious in your sketching.

For question two, I asked you to match some linear equations to their respective lines.

We should have matched line A there.

Y equals 2x minus a quarter as a positive gradient with a Y intercept zero, negative quarter.

That'd be just below the origin.

B and C should have been labelled as same.

They've got the same Y intercept, zero, 18, but B, Y equals 18 minus 2x should not be as steep as C, Y equals negative 5x plus 18.

That leaves us labelling that line as D, Y equals a quarter X.

Not very steep, but still a positive gradient.

Minus two.

A Y intercept of zero negative two.

That's the end of the lesson now.

In summary, we can recognise the determining features of the shape of a graph from its equation.

For example, we know that with an X exponent of one, the graph of the equation Y equals 8x plus 23 will be a straight line with a gradient of positive eight and a Y intercept of zero, 23.

Hope you've enjoyed this lesson as much as I have and I hope to see you soon for more mathematics.

Good bye for now.