Lesson video

Thank you for joining us in this lesson.

My name is Miss Davies and I'm going to help you as you work your way through this lesson.

There's some really exciting algebra coming up, so make sure that you've got everything you need and that you're really looking forward to getting stuck in.

Let's get started then.

Welcome to this lesson on checking and securing understanding of interpreting graphs.

By the end of this lesson you'll be really confident at reading, and interpreting points on a graph to solve problems. We're gonna look today at some linear relationships, and we're going to look at quadratic equations and quadratic graphs.

If you're not sure about the difference, pause the video and read those definitions.

We're also going to talk about parabolas.

If you haven't come across them before, a parabola is a curve.

It's got some special features.

The most important part is that they have a line of symmetry and that quadratic graphs are called parabolas.

Read through the formal definition if you need to now.

So, we're going to start by reviewing linear graphs.

We can plot the graphs of equations.

You've probably done this in a few different ways in the past.

One method is to use a table of values.

So, we're just going to do a little bit of a recap of how this works.

So if I wanted to draw the graph of y equals 2x, I can think of some x values, put them into a table, and then come up with some y values.

So because y has to be 2x, I just need to double my x values.

If I wanted to draw the line with equation y equals 4 minus 2x, I do exactly the same thing.

Come up with some x coordinates, and then I need y coordinates, which is 4 minus 2 lots of the x coordinate.

If you want to pause and check my answers, feel free to do so.

Then all we need to do is plot our points.

And they're both linear relationships, so they're both going to form a straight line.

Don't forget that when you draw your lines, they should go across the entire page because the relationship continues beyond our plotted points.

So Alex says, if we know the equation is linear, we do not need a whole table of values.

We can just use features of the graph instead.

Jun says, if I want to draw the graph of y equals 3x minus 6, I know the gradient is 3 and the y-intercept is 0, negative 6.

So we can use those features to plot a graph.

So if we plot the y-intercept, which for this graph is going to be negative 6, and then we need a gradient of 3.

And then we can draw in our line.

A gradient of 3 means we're stepping one right and three up to get to the next coordinate, one right and three up to get to the coordinate after that, and so on.

Let's think about this graph.

We've got 2x plus 3y equals 12.

How would you go about drawing that graph? Have a think.

So, Jun would like to rearrange it into the form y equals mx plus c, and then use the gradient and the y-incept.

Alex isn't so sure.

Alex says we only need two pairs of coordinates, so we could just find two coordinates that satisfy that relationship.

Of course, both methods are going to work.

You might want to pause the video and just have a think about what you would do.

So, let's have a look at rearranging.

So, if we subtract 2x from both sides, we've got 3y equals 12 minus 2x.

Divide by 3, we get negative 2 thirds x plus 4.

Now, we could absolutely use the y-intercept of 4 and a gradient of negative 2 thirds x to draw this line.

However, you might think that negative 2/3 x is a little bit of a difficult gradient to use.

So let's think about what Alex's method entails.

So, Alex says, we just need two coordinates that fit this rule.

Well, if we've got something in the form axe plus by equals c, there's two coordinates that's quite easy to find.

Any two coordinates would work, but we're going to look at the x-intercept and the y-intercept.

They're often the most efficient.

Because the x-intercept is where y is 0, if we substitute y as 0 into our equation, we can find the x-intercept.

So, we've got 2x plus 3 lots of 0 equals 12.

3 lots of 0 is of course 0, so 2x is 12, and x is 6.

So when y is 0, x is 6.

If we do the y-intercept as well, remember that's when x is 0.

So, we substitute x equals 0 into our equation.

3y is 12 and y is 4.

So when x is 0, y is 4.

We've now got two coordinates that we could plot and then draw in our line.

Let's have a look.

The coordinates are reasonably far apart, so it's going to be quite accurate to draw in a line between the two.

You might want to check that your gradient is constant all the way along that line.

Time to have a check then.

So, three Oak pupils plot in the line 2y plus 4x equals 4.

They're going to do it in some different ways.

So, Jun wants to use the gradient and the y-intercept.

He says the gradient is negative 4 and the y-intercept is 0, 4.

Can you spot what mistake he has made? Well done if you spotted that he has not made y the subject.

It still says 2y equals.

In order to find the gradient and the yincept, we need this in the form y equals mx plus c.

So if he divides every term by two, then he'll make y the subject, and then he'll be able to read off the gradient and the y-intercept.

Sam's going to do it a different way.

Sam wants to use a table of values.

Have a look at the purple box.

What has Sam done first to make this easier? And then can you fill in the three missing values? Off you go.

Let's see if we've got any of that mathematical language in here.

So, they have made y the subject.

They've written it in the form y equals mx plus c.

And that often makes it easier to use a table of values because you know how to find y given x.

Well, then, if you've got the missing values as 8, 6 and negative 4.

Remember, you can check that it's got a linear relationship to make sure you haven't made any mistakes.

Finally, Alex is going to use the x and y intercepts.

Complete their working to find the values for when x is 0 and y is 0, and then have a think about which of those three methods were your favourite.

Off you go! So, when y is 0, you should get x as 1.

And when x is 0, you get y as 2.

Well, I think all three methods were nice in this case.

The rearranging gave integer values, which was nice and easy to use.

And the x and y intercepts both had integer coordinates, so they're going to be nice and easy to use.

So, whichever method is your favourite is absolutely fine.

So, now we've reviewed how to plot graphs, we can look at how we can use them to solve problems. To start with, we can use graphs to solve equations.

So, here's the graph of y equals 8x minus 10.

Let's use it to solve 8x10 equals 2.

All we need to do is find a value for x, which makes this expression have a value of 2.

So, if we look for 2 on the y-axis, and often it's easier to draw in the line y equals 2.

And then we can read off by looking down to the x-axis and finding the corresponding value.

So, the solution to 8x minus 10 equals 2 is when x equals 1.

5.

If you wanted to check, you could substitute 1.

5 into the equation and see if it balances.

So, when a graph represents a scenario, we can actually interpret the coordinates in context.

It's really important that you think about what it is you're actually working out, not just following a process.

So, this is a conversion graph.

It converts degrees celsius into degrees fahrenheit.

It has an equation of F equals 1.

8 C plus 32.

Notice we've labelled our x-axis temperature in degrees C, and our y-axis temperature in degrees F.

So, what do you think about this one? What does the coordinate 0,32 mean in context? Can you write down or say out loud a sentence for what that coordinate is representing? Off you go.

Maybe you said something like this.

Zero degrees celsius is the same as 32 degrees fahrenheit.

That is what that coordinate represents.

So, Jun has a friend who lives in America.

She says that 60 degrees fahrenheit is about 20 degrees celsius.

How can we use the graph to check the accuracy of this statement? How would you do this one? Okay, there's a couple of ways to do this one.

What you could do is plot the coordinate 20, 60, which is what Jun says should be on our line.

However, we can see clearly that it's not on the graph.

So 20 degrees celsius is not 60 degrees fahrenheit.

It might be reasonably close.

Let's use another method to check how close.

So if you actually look for 60 degrees fahrenheit and then draw a vertical line down to your x-axis, you can see that it's a lot nearer to 15 degrees than it is to 20 degrees.

There's actually quite a big difference, isn't there, between 15 degrees C and 20 degrees C.

So, Jun's friend probably should have said, it's roughly 15 degrees C.

And then we'd have a better understanding of what 60 degrees fahrenheit is actually like.

Time for you to have a go.

I'd like you to use the conversion graph to make these conversions.

Off you go.

Let's have a look.

10 degrees C should be 50 degrees fahrenheit.

And 5 degrees fahrenheit should be negative 15 degrees C.

Just check the graph if you didn't get values close to those.

Right, time to put those skills into practise.

For our first question, I'd like you to use this graph to solve the following equations.

Off you go and come back when you're ready for the next set.

Well done.

So, this second set is in context.

So, Lucas volunteers at local playgroup.

For his homework, he drew a distance time graph of a race that two children ran.

You can see on the right hand side our distance time graph.

I'd like you to answer these questions for the graph that Lucas has drawn.

Make sure you're thinking about the context and writing your answers in full sentences where necessary.

Off you go.

Question three.

So, a plumber calculates the amount to charge a customer using this linear relationship.

It includes a callout fee and a price per hour.

I'd like you to think about that context when you answer these four questions.

Off you go, come back when you're ready for the answers.

So for the first one, you should have x is 3, then x is 0, then x is negative 2.

4.

Make sure you're happy with reading off decimal values on a graph.

And then dx is negative 3.

So, let's think about this context now then.

So, child 1 ran 4 metres in 2 seconds.

If you you just wrote four you haven't answered the question properly, so four metres.

For b should be three seconds, and c I wonder if you noticed that child two started further ahead, so the y-intercept is saying that zero seconds, so before the race has started, the child is already two metres.

So that probably represents a two metre head start in this context.

5,10 and 8,10.

So 5,10 means that child one took five seconds to run 10 metres.

And 8,10 says that child two took 8 seconds to run 10 metres.

10 metres is probably the length of the race.

Now, you could have combined that with your previous answer and said that actually Child 2 didn't run 10 metres.

They actually only ran 8 metres.

But to get 10 metres away from where they should have started, it took them 8 seconds.

And the last one, there's some different answers you could have come up with here.

You might have said something like, they don't run at a steady pace.

Normally you'd accelerate at the beginning, and then slow down later on, whereas a straight line graph suggests you're at the same speed all the way through.

For a plumber, 0.

80 suggests that a job that takes zero hours costs £80.

You might have said that this was a call out fee.

For B, you should have £120.

For C, 10 hours.

And then the price per hour is actually the gradient of this line.

If you think about every hour you add on, how much does the price go up? We can see from the line that if you add on two hours, the price goes up £20.

So for every hour, it's £10.

So, £10 per hour.

Well done.

We're now going to have a go at reviewing our quadratic graphs.

So, we can use a table of values to plot quadratic graphs too.

Let's have a look at y equals x squared.

So if we square our x values, remembering that when we square a negative value, we get a positive value, and then we can plot our values on a graph.

Of course, this is not a linear equation, so it does not form a linear graph.

Therefore, we cannot draw a straight line.

We can see that really clearly.

Y equals x squared is a quadratic equation.

Its graph is called a parabola.

And what we can do is draw a curve instead.

Jacob has noticed that we've got a line of symmetry, and in this case it's the y-axis.

Okay, let's have a go with some trickier quadratic equations.

We're going to do y equals 2x squared minus 4x.

But we do exactly the same thing, we just need to substitute the required values for x.

I would suggest that you start with zero and then the positive values, and then go back to the negative values.

The positive values are generally easier.

And then you can look at some patterns to make sure you get the right values for the negative x.

If you're using a calculator, make sure that you're putting brackets around your substituted value.

Right, we've done the first three.

Sam says, have we done something wrong? The y-coordinate is zero again.

What do you think? No, our working out is perfect.

It's just that there is a line of symmetry somewhere on a parabola.

Therefore, it is common for there to be repeated values in the table.

It doesn't always happen, but it is usual with a quadratic graph to see repeated values in the table.

Let's finish off our positive values.

Let's have a think now.

What do we expect x equals negative two and x equals negative one to produce? See if you use Jacob's idea about symmetry.

Okay, let's try it out.

We should have 6 and 16.

I wonder if that's what you predicted looking at those values.

And you can see we have got a line of symmetry because the values are repeated either side of when x is 1.

Alex is having a go at plotting this one.

I think when x is negative 1, the y value will be 4 because parabolas have a line of symmetry.

I'd like you to substitute into this equation to see if Alex is correct.

Off you go.

Right, whether you're doing this by hand or using a calculator, you need to make sure you put brackets around the value you're substituting.

So, we should have negative one all squared plus two lots of negative one plus one.

Negative one squared is one, so we've got one, add negative two, add one, and that gives us zero.

So, Alex is incorrect.

There is going to be a line of symmetry somewhere, but it's not always the y-axis.

So there could be a line of symmetry somewhere else in the table, or there might not even be in the table of values.

It might be somewhere else on the graph.

So like with linear equations, we can solve quadratic equations using their graph.

So, let's do x squared plus 4x plus 1 equals 6.

First, we can draw the graph.

And then we want to know when this is equal to 6.

If you draw the line y equals 6, you'll see that we have two solutions.

Often when you have quadratic equations, there are two solutions, but not always.

Our two solutions in this case are x equals negative 5 and x equals 1.

Like we just said, there could be two solutions, or there could be one solution or no solutions.

Let's have a look at some examples.

So, here is the graph of y equals 2x squared minus 4x plus 1.

If I want to know when that equals 7, you'll see we have two solutions.

When that equals 3, again we have two solutions.

We might not be able to tell exactly what they are because they're between integer coordinates.

But we do have two solutions and they're approximately negative 0.

4 and 2.

4.

So, where on the graph is that only one solution? Can you write an equation that has one solution? See if you can do this.

Well, if you spotted it's at that very bottom point of the curve, it's where x is 1 and y is negative 1.

What that means is the equation 2x squared minus 4x plus 1 equals negative 1 only has one solution.

Can you think of an equation then that will have no solutions? Okay, it's going to be anywhere below that bottom point of the curve.

So for example, 2x squared minus 4x plus 1 equals negative 2.

Or essentially, that expression equal to any value less than negative 1.

And it's that point of the graph underneath that bottom point of the curve.

We can add a context, just like we did with linear graphs.

So, this is the horizontal and vertical distance of a shot put throw.

Why do you think the graph stops when we get to 20, 0? Okay, well that's going to be where the shot put lands on the ground again, because it can't go lower than 0 metres above the ground, unless it's going through the ground, of course.

If we want to know the horizontal distance when the shot put is a height of three, we find a height of three.

And then we read down to find our horizontal distances.

The challenge with these questions is always reading the context and understanding what the graph is showing you.

What is the highest point that the shot put reaches? Can you find it? So, the highest point is four metres, making sure you're reading off the right value.

Quick check then.

So, this parabola models an athlete completing a pole vault jump.

You've got horizontal and vertical distance.

What was the highest point reached by the athlete? Off you go.

Out of those options it has to be 6.

25.

Right, how far has the athlete travelled horizontally when they are four metres above the ground? What do you think? Okay, so we need to find four metres above the ground.

And you'll see that there's two solutions, one when the horizontal distance is 0.

2, then obviously on their way back down again, when the horizontal distance is 0.

8.

Time for you to have a go.

I'd like you to use the table of values to plot the parabola with equation x squared minus 2x minus 3, and then use it to find the solution to those equations.

Off you go.

For question two, I would like you to read the context carefully, and then use that to answer the questions, making sure you're thinking about units and full sentences where necessary.

Off you go.

And this time we've got an Olympic diver.

Again, make sure you read the graph carefully and then answer the questions.

Off you go.

Let's have a look at our answers then.

I've used graphing software, so my graph is perfect.

However, if you plotted this by hand, it might not be exactly the same as mine.

What you want to do is check for key coordinates.

Does it go through the coordinate negative 2, 5? Does it go through the coordinate 0, negative 3? Does it have a lowest value at 1, negative 4? And does it have that line of symmetry? Use the table of values to check you've got the right points.

Your solutions then, roughly negative 1.

8 and 3.

8.

Again, that's going to depend a little bit on how your graph is drawn.

You should get exactly negative 1 and 3.

And you can see that from the table of values as well.

You can see we've got negative 1, 0 and 3, 0.

And you've got one solution of x equals 1 for that last one.

And again, you can see that in your table of values.

For our penguin, then, can you make sure you've checked that your table of values is correct? And again, you should have a nice, smooth curve for your parabola.

You should have a line of symmetry when x equals 4.

Why did the table of values start at zero? Well that means zero minutes, so that's when the penguin is starting the dive.

It does not make sense to have negative values for minutes.

The deepest point that the penguin got to was 160 metres.

It was underwater for eight minutes and then 70 metres below sea level was after one minute, and then they returned again after seven minutes.

Penguins can actually dive under the water for a really long time.

And some emperor penguins can actually get to a depth of 300 metres.

So our Olympic diver, well done if you spotted that the person writing that article has not understood what the graph is about.

The x-axis is representing horizontal distance, not time.

There's no way to gather any information about time from this graph.

It's not uncommon for graphs to be incorrect, or for people to misinterpret graphs.

So, it is something that's actually worth keeping your eye out for when you're watching other people working with graphs.

Something like 11 metres was the highest point the diver reached and they had moved 0.

2 metres horizontally to reach this height.

If you think about diving off a diving board, you're obviously going to have to go forwards as well to clear the diving board before you start falling back towards the water.

See, 0.

410 is the moment where the diver was in line with the starting point again.

So like we said, they need to have cleared the diving board, so it has moved 0.

4 metres horizontally and is back at the same height as where they jumped from.

Well, I hope you feel like you're really secure now in your understanding of graphs and that you're going to be able to use those going forward.

We've looked at plotting linear graphs, and the different ways that you might want to go about doing that including a table of values and this idea of gradient and y-intercept.

We can use linear graphs to find solutions to equations, and we can put those in context as well and find solutions to problems. We can do exactly the same with linear graphs as with quadratic graphs.

Just remembering that they're not gonna form a straight line, so you have to be able to draw a curve as neatly as you can through the points, or use graphing software and technology to get it perfect.

Remember as well that we can have no solutions, one solution or two solutions from a quadratic equation, and you've seen how we can tell that from the shape of the graph.

And then we had some fun playing around with some different real life scenarios, some different Olympic sports, and we found out some fun stuff about penguins.

Thank you for joining me in today's lesson and I look forward to seeing you again.