Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Today we're checking and securing our understanding of drawing vertical and horizontal graphs.

Vertical and horizontal lines are really, really useful in mathematics, so let's find out what it's all about.

Our learning outcome is what we'll be able to draw graphs of the form y equals a, x equals a and interpret linear graphs in context.

Some keywords that we're gonna hear throughout the lesson.

Linear, gradient, and intercept.

The relationship between two variables is linear if when plotted on a pair of axes, a straight line is formed.

The gradient is a measure of how steep a line is, and an intercept is the coordinate where a line or curve meets a given axis.

Watch out for these words throughout this lesson.

Two parts to our learning today.

We're gonna start by plotting vertical and horizontal lines.

We can plot a linear graph from its equation using a table of values.

For the equation y equals x plus four, we can substitute those x values in to find the corresponding y value.

When x equals zero, y equals zero plus four.

That's four.

When x equals one, y equals one plus four.

That's five.

When x equals two, y equals two plus four.

That's six.

And when x equals three, y equals three plus four.

That's seven.

Are you starting to spot a pattern in these y values? That pattern enables you to find the rest of the y values and from here we can plot these points to draw the graph.

There are the coordinates and a linear graph of y equals x plus four.

Y equals x plus four is an equation in the form y equals mx plus c.

In the form y equals mx plus c, it's the coefficient of x, our m value, which tells us the gradient of our line.

C is the constant and it tells us the y-intercept, or rather, it tells us the y coordinate at the y-intercept.

So in the case of y equal x plus four, that's a coefficient of one, i.

e.

, it's 1x.

That gives us a gradient of one, which I can see on the graph.

A gradient of one.

The constant four tells us the y coordinate four at the y-intercept, so our y-intercept is zero, four.

If that's the line y equals 1x plus four, how do you think the line y equals 0x plus four will look? Pause this video and have a think about that now.

Welcome back.

I wonder what you think.

Lots of ways we could think about it.

We could do a table of values.

Y equals 0x plus four.

Let's find some values and let's plot it.

When x equals zero, y equals zero lots of zero plus four.

That's four.

When x equals one, y equals zero lots of one plus four.

That's four.

When x equals two, y equals zero lots of two plus four.

That's four.

Are you starting to see a pattern? What do you think is gonna happen when x equals three? Well done, y is going to equal four.

If we continue and populate that table, we get this, and it's unusual by contrast to most tables we see when plotting linear equations.

Because of the unique multiplicative property of zero, it doesn't matter how much x varies in the equation y equal 0x plus four, y will never vary.

It will always be four.

From here, we can plot those coordinates and draw a line through them just as you would for any linear equation.

The line y equals 0x plus four, the x coefficient is zero, so we can call this a gradient of zero.

The constant four tells us there's a y-intercept at zero, four.

A gradient of zero means no change in y.

Hence, we see a horizontal line.

That is a zero gradient.

Y equals 0x plus four as an equation simplifies to y equals four.

We call this horizontal line y equals four.

Every single pair of coordinates has a y value of four.

Obviously all the coordinates we pick out of our table of values, you can see the y value of four.

I could pick 0.

5, four.

It's a y coordinate of four.

I could pick negative 1.

25, four.

That's a y coordinate of four.

I could pick eight over three, four, and I still get a y coordinate of four.

Hence, this line is called y equals four.

Where would the line y equals 0x plus one be on this graph? Pause and have a think about that now.

Welcome back.

I wonder what you thought.

I wonder how you approached this.

There's lots of ways we can think about it.

We could think of it, it's got a y-intercept EP of zero, one or a y-intercept at zero, one and a gradient of zero.

In that case, we can plot that y-intercept at zero, one and draw our zero gradient, a horizontal line.

There it is, the line y equals 0x plus one.

Alternatively, you could have generated a table of values and you would've found all those coordinates and drawn the exact same horizontal line.

Funnily enough, this line is called y equals one.

Another check to see that you've got this.

I'd like you to identify at least three coordinate pairs on this line.

Hence, find the equation of the line.

Pause.

Pick any three coordinate pairs.

Tell me that equation.

Welcome back.

I'm gonna pick some coordinate pairs out.

One, negative three, two, negative three, three, negative three.

They were just three obvious coordinate pairs that jumped out to me, but I could've picked any.

I could've gone for negative 0.

5, negative three, negative two and a quarter, negative three.

It didn't matter which ones I picked.

You should've noticed this.

Every single y coordinate has a value of negative three.

Hence, this line is called y equals negative three.

Another check now, and it looks an awful lot like the last check, but there's some things special and unique about this case, so I want us to have a look at it.

I'd like you to identify at least three coordinate pairs on this line.

Hence, find the equation of the line.

Pause and do that now.

Welcome back.

I wonder what we discovered.

Three coordinate pairs.

I could go for one, zero, two, zero, three, zero.

You could go for anything.

You could go for negative one, zero, negative two, zero.

But whichever ones you picked, you always found this.

Every single y coordinate has a value of zero.

Hence, this line is called y equals zero.

Did you notice it's also the x-axis? You would be familiar with that phrase.

That line is the x-axis.

But did you know that we can also call the x-axis y equals zero? That's going to be useful in your future learning.

In the form y equals mx plus c, the line y equals 0x has a gradient of zero.

The line y equals 1x has a gradient of one.

Y equals 2x has a gradient of two.

Y equals 5x has a gradient of five.

Y equals 10x has a gradient of 10.

Y equals 100x has a gradient of 100.

And you see those lines getting steeper and steeper and steeper.

How do you think we define the gradient of a perfectly vertical line? Pause and have a think about that.

Welcome back.

I wonder what you thought.

A perfectly vertical line.

If we try to define a vertical line using the form y equals mx plus c, we come across a problem.

Gradient, when we calculate it, we use this formula.

It's the change in y over the change in x.

The problem we have is y is infinitely changing, but x is not changing at all.

If there's no change at all in x, we have to write zero in the denominator position.

That's a problem.

Dividing by zero is undefined.

Hence, this gradient is undefined.

It's a vertical line, which means it also has no y-intercept.

This line cannot be defined using the form y equals mx plus c.

M is the gradient, which is undefined.

C helps us to identify the y-intercept, which we don't have.

So we can't use the form y equals mx plus c.

The good news is we can use the form axe plus by equals c to define a vertical line.

This line is 1x plus 0y equals two.

I can justify that to you by picking out some coordinates and showing you that when I substitute in x equals two, y equals one, the equation is satisfied.

Pick another coordinate, two, four.

When x equals two and y equals four, the equation is again satisfied.

One in the fourth quadrant, two, negative two.

Substitute in that x value and that y value.

The equation is again satisfied.

This is definitely the line 1x plus 0y equals two.

No matter how much y varies, x is always two.

Hence, this line is called x equals two.

If we were to simplify the equation 1x plus 0y equals two, well, y multiplied by zero is always going to give us zero, so that's eliminated, and 1x, well, we don't write 1x.

We just call that x.

Hence, x equals two.

Another way you know this is every single pair of coordinates has an x value of two.

You could keep going picking coordinates off that line.

Your x coordinate will always be two.

Hence, the line is called x equals two.

Quick check you've got that.

I'd like you to identify at least three coordinate pairs on this line, hence, find the equation of the line.

Pause and do that now.

Welcome back and see how we did.

I picked out these three coordinates.

You could've picked out any coordinate pairs you liked.

I went for negative one, three, negative one, four, negative one, negative two.

But whichever ones you picked out, there was no escaping the fact that every x coordinate has a value of negative one.

So what do you think we call this line? Well done.

We call it x equals negative one.

Another check now.

Andeep says this line is parallel to the y-axis and goes through three, so this line must be y equals three.

A common misconception this, but how would you explain to Andeep why he is wrong? Pause and think about that explanation now.

Welcome back.

Lots of ways we can think about this and unpick this misconception.

You might have said lines in the form x equals a give us a vertical line, so this must be the line x equals three.

Alternatively, you might have demonstrated by picking a few coordinate pairs from the line, the coordinates three, two, three, zero, three, negative four.

Well, it's not the case that y equals three.

I can see y being equal to two, y being equal to zero, y being equal to negative four.

It is in fact x that is equal to three.

Every x coordinate we pick is a value of three.

You might have demonstrated to Andeep that that is the case.

Practise time now.

Question one.

I'd like you to plot the lines x equals five, x equals negative five, y equals negative four, and y equals 2.

5 on the axie.

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

I'd like to identify the equations of those lines.

Pause and do those problems now.

Question three.

Slightly trickier, this, but very achievable.

You'll be all right.

What is the area of the rectangle bound by the lines x equals 35, x equals negative 12, y equals 27, and y equals 15? Lovely problem, this.

Enjoy.

Feedback time now.

Let's see how we got on with question one.

Line x equals five will be there.

For x equals negative five, negative five, zero will be on that line and negative five, five will be on that line.

There's the straight line x equals negative five.

For y equals negative four, zero, negative four will be on that line, four, negative four will be on that line, or any two coordinate pairs whereby the y coordinate is negative four.

You'll get that straight line.

There's y equals negative four.

For y equals 2.

5, zero, 2.

5 is on the line, 2.

5, 2.

5 is on the line.

We get that horizontal line y equals 2.

5.

So the lines are x equals five, x equals negative five, y equals negative four, y equals 2.

5.

You might wanna pause now and just check that your lines are in exactly the same place as mine.

Question two, I asked you to identify the equations of the lines on this graph, and you should have got x equals three for that vertical line, x equals negative four for that vertical line, y equals six for that horizontal line, and y equals negative five for that horizontal line.

Question three.

Delightful problem, this.

We're asked to calculate the area of a rectangle.

This problem can be tackled in a lot of ways.

A visual representation of the problem will definitely help us.

I can visually represent those four lines with a sketch like this.

We can look at the base of this rectangle and consider that base sits between x equals 35 and x equals negative 12.

35 minus negative 12 is 47.

That length there is 47.

How about the height? Well, that's the line y equals 15 and the line y equals 27, so 27 minus 15 will tell us that height is 12.

From there, we use the formula for every rectangle, base times height.

47 multiplied by 12 is 564, so the answer is 564 square units.

On to the second part of the lesson now, using vertical and horizontal lines.

There's lots of uses of vertical and horizontal lines in mathematics.

One is in solving equations graphically.

When we're asked to solve 4x minus seven equals nine, what we're actually being asked is when does the line y equals 4x minus seven intersect the line y equals nine? We can see the line 4x minus seven and the line y equals nine.

Where do they intersect? At that coordinate there.

At that moment, x is equal to four.

We can see that by reading down a vertical line.

X equals four is the only value which satisfies the equation.

We can calculate that numerically, but we can also see it with our vertical and horizontal lines on this graph.

We can use vertical and horizontal lines to solve simultaneous equations graphically, too.

If we're asked to solve this pair of simultaneous equations, what we're actually being asked is when does the line 2x plus 3y equals 15 intersect the line 5x plus 4y equals 13? There's the two lines and the intersection.

From there we can read vertically down to the x-axis to find when x equals negative three and horizontally across to the y-axis to find when y equals seven.

X equals negative three, y equals seven are the only values which satisfy this pair of equations simultaneously, and we needed horizontal and vertical lines to find those values.

Quick check.

You've got this now.

I'd like you to use the graph to solve the equation, the equation being five minus 4x equals minus three.

Pause.

Tell the person next to you or say aloud to me at the screen, what's the solution to that equation? Welcome back.

I do hope you spotted the point of intersection and then used vertical line to find it's when x equals two.

Our solution is x equals two.

Another check now.

Jacob is using this graph to solve a pair of simultaneous equations.

The solutions are x equals negative two and y equals eight, Jacob says.

What error has Jacob made? Pause.

See if you can spot it.

Welcome back.

How did we get on? Let's find out.

You might have said vertical lines have the form x equals a, so this is the line x equals eight.

You might have said horizontal lines have the form y equals a, so this is the line y equals negative two.

Hence, the solution is x equals eight and y equals negative two.

It's not just in graphs of equations that vertical and horizontal lines are useful.

This is a conversion graph between GBP and USD.

That's Great British pounds and United States dollars.

We can convert 100 pounds into US dollars by drawing a vertical line up to our conversion line, like so, and a horizontal line across.

This enables us to convert 100 British pounds into 125 US dollars.

The solution should be compared to the context to check for misreads.

If we are asked to convert 100 pounds into US dollars, we might accidentally read from the wrong axis.

We might accidentally read horizontally across, vertically down, and it would appear as if 100 pounds converts to 80 US dollars.

We know this is wrong if we look at the initial exchange rate.

We saw earlier that one pound is equal to $1.

25.

This will remain proportionately true throughout.

Every pound will get you $1.

25.

So when converting, the dollar value is always gonna be larger than the number of pounds.

When we look at our answer, 100 pounds is $80, we know it's wrong.

That dollar value should be bigger than the pound value.

Quick check you've got that.

Jacob is reading from an inches to feet conversion graph to find out how many inches are in six feet, but there are no labels on the axes.

"Those lines say six feet converts to 0.

5 inches," says Jacob.

Jacob knows immediately this is wrong.

How does he know? Pause.

How would you answer that? Welcome back.

So we misread the graph and thought that six feet might convert to 0.

5 inches.

Jacob had a wonderful idea.

He looked at the ruler in his pencil case and said, "The ruler in my pencil case tells me there's 12 inches in just one foot, so this cannot be right." Nice idea, Jacob.

Indeed, we have to go across to the six on the horizontal axis, read up vertically to the conversion line across horizontally, and we see six feet must be 72 inches.

Asking yourself, "Does that answer feel sensible for this context?" can help you to spot errors.

Of course, if the graph's axes had been labelled, we'd have been less likely to have had this problem in the first place.

Another quick check that you can spot the sensible answer in this context.

A taxi firm charges a fixed fee per journey, then a cost per mile.

This can be graphed, but the labels have been left off these axes.

How much is a journey of eight miles? So which is it? Is an eight-mile journey one pound or is an eight-mile journey 29 pounds? Pause.

Have a think about that.

Welcome back.

I hope you said it's option B, 29 pounds, because consider the context.

No taxi firm's gonna charge one pound for an eight-mile journey.

Additionally, if we label those axes, you see miles on the horizontal axis, pounds on the vertical axis.

There's the coordinate zero, five.

That means zero miles costs five pounds.

That's the fixed fee.

If it were the other way round, then the taxi company would be taking you five miles for no pounds.

I don't think so.

Practise time now.

And for question one, I'd like you to use this graph to solve these three equations.

Pause.

Use the graph to pick out your solutions now.

Question two.

I'd like you to use this graph to solve these pairs of simultaneous equations.

Pause and pick out your solutions to those three now.

Question three.

A taxi firm has a fixed fee per journey, than a cost per mile.

Part A, I'd like you to label the axes.

Part B and C require you to do some reading from your graph.

Pause and give this problem a go now.

Feedback time now.

Question one was using these graphs to solve equations.

For A, we were looking at that intersection where the line y equals five minus x intersects the line y equals seven.

We need to read vertically down to find that x value of negative two.

Solution to part A.

It's x equals negative two.

Part B.

We are interested in that intersection where the line y equals seven intersects the line y equals 4x minus five.

Using a vertical line, we can read down to find the x solution, x equals three.

For part C, we want that intersection where y equals 4x minus five intersects y equals five minus x.

Vertical line to help us read the x value.

Our solution x equals two.

For question two, we're using the graph to solve pairs of simultaneous equations.

For part A, we are interested in that intersect, and we need a vertical line to read the x value and a horizontal line to read the y value.

Our solution x equals negative three, y equals negative three.

For part B, we're interested in that intersection.

A vertical line to read the x value.

A horizontal line to read the y value.

Our solution x equals seven, y equals two.

For part C it's this intersection we're interested in.

A vertical line for the x value, horizontal line for the y value, giving us a solution x equals one, y equals nine.

For question three part A, we were asked to correctly label the axes.

I hope you labelled the horizontal axis with our distance, miles, and our vertical axis with our cost in pounds.

It had to be this way around.

If you look at the coordinate zero, eight, that's zero miles costing us eight pounds.

That's the fixed fee, eight pounds, before you've even travelled anywhere.

If we'd accidentally labelled it the other way round, for zero pounds, we'd be travelling eight miles.

Not gonna happen, so it must be this way round.

For part B, use the graph to state how much a five-mile journey costs.

Let's read up with a vertical line from five miles and a horizontal line across to the cost axis.

It's 18 pounds.

For part C, how far could we travel for 20 pounds? We'll read horizontally across from 20 pounds.

It's that point there.

And then down to six miles.

For 20 pound, we could travel six miles.

It's the end of the lesson now, sadly, but we've learned that we can draw graphs of the form x equals a and they form vertical lines.

We can draw graphs in the form of y equals a and they form horizontal lines.

Vertical and horizontal lines interpreted in context enable us to solve a variety of problems in mathematics, such as the solving of equations and the reading of conversion graphs.

I hope you've enjoyed this lesson as much as I have, and I'll see you again soon for more mathematics.

Goodbye for now.