Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Lovely to see you again.

Today, we're drawing reciprocal graphs.

Trust me, they're gorgeous.

You're gonna like this one.

Our learning outcome is that we'll be able to generate coordinate pairs for a reciprocal graph from its equation and then draw the graph.

Reciprocal will be a keyword today.

The reciprocal is the multiplicative inverse of any non-zero number.

Any non-zero number multiplied by its reciprocal is equal to one.

You'll see a lot of reciprocals today.

Two parts to our learning in this lesson.

We're gonna begin by drawing reciprocal graphs.

Maths can be used to model many real-life situations.

A crate of water contains 24 bottles.

How many bottles are there in two crates, four crates, eight crates, or c crates? Hmm.

Last one seemed unusual.

See if you can figure those out.

Pause and do that now.

Welcome back.

Hopefully, you said there'll be 24 by two equals 48 bottles in two crates.

There'll be 96 bottles in four crates, 192 bottles in eight crates.

And then for the last one, there's nothing unusual about it.

We just need to take the number of crates and multiply it by 24 and we'll get the number of bottles.

We could write an equation for what's happening here.

When we graph this situation, we see a direct proportion graph like so.

This is the graph of b equals 24c.

Crates is along the horizontal axis and bottles is the vertical axis.

You'll notice it's a straight line graph with a gradient of 24.

You also notice it has a y-intercept at the origin 0, 0.

You also see constant multiplicative relationships.

What's meant by that? Well, if I pick any two coordinate pairs out, there's a multiplicative relationship between them.

If I take five crates and double it to 10 crates, I take my 120 bottles and double it to find 240 bottles as constant multiplicative relationships all over this graph.

It's because all the coordinate pairs are linked by the constant 24.

Let's keep the 24, but change the nature of the real-life situation and see how it affects our model.

If we said a wall takes one person 24 hours to paint, how many hours would it take two people, four people, eight people, or p people where the number of people varies? Pause and work out those four now.

Welcome back.

Hopefully, you spotted on this occasion it's about division.

We're gonna take the 24 hours and divide it across those two people.

We're gonna get 12 hours of work.

24 divided by four means it'll take four people six hours.

24 divided by eight means it'll take eight people three hours.

We're taking 24 dividing it by the number of people and that's how many hours the job will take.

This situation forms a graph of inverse proportion.

This time, we've got people on the horizontal axis and hours on the vertical axis.

There's our coordinate pairs.

That's two people taking 12 hours, four people taking six hours, eight people taking three hours.

This is the graph of h equals 24 divided by p.

It's an inverse proportion graph.

Inverse proportion graphs have the form y equals k over x, where y and x are variables and k is a constant.

They're non-linear.

They intercept neither axis.

The graph intercepts neither axis.

Let's have a bit more of a think about that.

If p equaled zero, we'd have 24 divided by zero.

Dividing by zero is undefined.

Try typing it into your calculator, it'll confirm it for you.

Additionally, there's no p value we can take that makes h equal zero.

You could keep adding more and more and more people and it'll never take zero hours.

We could have a thousand people and we'll still have 24 thousandths as a h value.

It'll never reach zero.

That's why this graph intercepts neither axis.

So the graph intercepts neither axis, it's non-linear and it's full of reciprocal multiplicative relationships.

Let's have a closer look at that.

Reciprocal multiplicative relationships, unlike our direct proportion graph where there are constant multiplicative relationships, something different's happening here.

If we compare the coordinate pair 2, 12, to the coordinate pair 4, 6, what's happened? We've doubled the number of people, therefore, the number of hours has halved, have a look, to a half.

They are reciprocals of one another.

What if we quadrupled the number of people? We'd go from having two people to having eight people? It takes a quarter of the time.

Look, the coordinate pairs 2, 12 and 8, 3 are linked by this reciprocal multiplicative relationship.

Four and a quarter are reciprocals of one another.

Quick check you've got this.

Which statements are true for the graph of this situation? Some pupils are sharing 500 grammes of cookie dough.

Is A, B or C true? Pause and decide now.

Welcome back.

I wonder what you said.

Hopefully, you said that A is not true.

It will not form a linear graph.

Hopefully, you said B is true.

The graph will intercept neither axis.

There's our graph.

And hopefully you said C is true.

There'll be reciprocal multiplicative relationships.

Let's explore this.

We'd have one person getting all 500 grammes of cookie dough, lucky them.

Or if we pick any other coordinate pair from the graph, 10 people would end up with 50 grammes each.

When we compare those coordinate pairs, we have 10 times the number of people.

They therefore get one 10th the amount of cookie dough, what a shame.

10 and 1/10th are reciprocals.

There are reciprocal multiplicative relationships all over that graph.

What happens if we remove the context? We already know the graph h equals 24 over p appears in the first quadrant, but will it appear in any other quadrant if we model abstractly, if you will? Will any coordinate pairs appear in the fourth quadrant? Will we ever have a moment where p is positive and h is negative? What do you think? Absolutely not.

A positive value, in this case our constant is 24, divided by another positive will make a positive.

We won't get any coordinate pairs plotted in the fourth quadrant.

What will happen when we substitute in negative p values? In what quadrant will those coordinate pairs appear? I'm gonna leave you to figure this out for yourself.

What would happen if p were negative two or p were negative four or p were negative eight? Where would those coordinate pairs appear on the graph? Pause and have a think Welcome back.

We can test a few values to see where those coordinate pairs would end up.

If p were negative 2, 24 divided by negative two equals negative 12.

That coordinate appears there.

24 divided by negative four equals negative six.

So if p equals negative four, h equals negative six, there we go.

24 divided by negative eight equals negative three, If p is negative eight, h is negative three.

What if p were negative 24? h would be negative 21.

That coordinate pair appears there.

A positive value, in this case the constant 24, divided by a negative is always a negative result.

Hence, those coordinates are in the third quadrant.

This is the graph h equals 24 over p.

A positive value divided by a positive value always results in a positive value, and a positive value divided by a negative value always results in a negative value.

The graph only exists in quadrants one and three.

Where can you see a multiplicative relationship? Let's pick any coordinate to start.

2, 12.

Let's compare it to the coordinate pair 12, 2.

We have to multiply p by six, therefore h becomes 1/6th, the size, reciprocals.

What if we compare the coordinate pair 12, 2 to the coordinate pair -6, -4? Well, on the left-hand side we have to multiply our p value by negative 1/2.

What's gonna happen on the right-hand side? Of course, we had to multiply by negative two.

Even though I've picked coordinate pairs from different quadrants, we've still got this reciprocal multiplicative relationship.

Negative 1/2 and negative two are reciprocals of one another.

Let's compare the coordinate pair -6, -4 to -24, -1.

We see the same reciprocal multiplicative relationship.

These are all reciprocal relationships, therefore, this is an example of a reciprocal graph.

Quick check you've got this.

I'd like you to find the missing values in this table of values for the reciprocal graph y equals 15 over x.

Pause, fill in those missing spaces now.

Welcome back.

Hopefully, 15 divided by three equals five, and you spot 15 divided by five equals three.

We can put those two simple positive values into our table.

Let's do a couple of the trickier ones.

15 divided by negative one.

Remembering that's negative 15.

A positive divided by a negative, we're going to have a negative result.

Same thing happens for 15 divided by negative 15.

We get a negative result, this time negative one.

Pop those values into our table.

In which quadrants will this graph appear? Now that we've got a completed table of values for the graph y equals 15 over x, well, we see those coordinate pairs plotted in quadrant one, quadrant two, quadrant three or quadrant four.

Pause and have a think about that now.

Welcome back.

I hope you said quadrant one because we'll have positive x values and positive y values.

It's not quadrant two, it is quadrant three.

Negative x values will give us negative y values, and we'll only see the graph in the two quadrants, one and three.

We won't see it in quadrant four.

There is our graph.

The reciprocal graph, y equals 15 over x.

Jun and Jacob are completing a table of values for the positive x values of the reciprocal graph y equals one over x.

Jun fills those values in and then says, "Some of these decimals are getting awkward.

Have you got any advice?" That's awesome, Jun, it's good to ask your peers for advice in mathematics sometimes.

Jacob says, "Leave your answer as a fraction.

Try it and see what happens." Jun starts to populate the table with these values.

Jacob says, "Well done, Jun.

This is much more efficient.

The table is much quicker to do this way." Jun says, "That was a lot easier.

The y values are always the reciprocal of the x values, so fractional form is brilliant." I couldn't agree more, Jun.

Jun and Jacob now plot the graph of y equals one over x using their table of values.

The coordinates look like so.

Jun asks, "Does the graph stop here?" Jacob says, "No, you need to consider what will happen when x is between zero and one." I'm gonna leave this down to you to figure out.

What do you think happens when x is greater than zero or less than one? Pick any numbers between those two values and see if you can find some coordinate pairs and predict what happens to our graph.

Pause and have a think about that now.

Welcome back.

You could test any values between zero and one for x.

I'm gonna to use x equals 1/2, x equals 1/3, x equals 1/4 and x equals 1/5th.

I'll get those y values.

Jun spotted, "Of course, the same pairs of reciprocals." When we plot those, they end up there making our graph look like that in the first quadrant.

Jun then asks, "Aren't the negative x values really difficult to plot?" Jacob says, "No, we just need to look at a few values and you'll spot something." Let's do that then.

What about when x equals negative 10, negative two, negative one, negative 1/2 and negative 1/10th.

We get those respective y values, the negative x value giving us a negative y value.

And when we plot them we notice we get the exact same shape of graph, just this time in quadrant three instead of quadrant one.

This is the reciprocal graph y equals one over x.

Quick check you've got this now.

Which of these coordinate pairs will be on the graph of y equals one over x? Four pairs to pick from.

Some of them will be on the graph, some of them won't be on the graph.

I'll leave you to decide.

Pause and have a good think now.

Welcome back.

I hope you said A, 1/50 and 50 will be, it's because they're a pair of reciprocals.

Multiply one over 50 by 50, you get one.

They're a pair of reciprocals.

The same thing is true of coordinate pair B.

That's a pair of reciprocals.

C will not be on the graph.

They're not reciprocals, they're negative reciprocals.

Multiply that x and y value together, you'll get negative one, not positive one.

That will not be on this graph.

D will be on the graph, negative 1/5th, negative 5, they are reciprocals.

That'll be on the graph in the third quadrant.

Reciprocal graphs can have the form y equals k over x.

We've seen y equals one over x.

In this example, we're gonna look at y equals 10 over x.

We could pick some coordinate pairs and draw the graph like so.

Jun says, "So reciprocal graphs only ever appear in the first and third quadrants." Well, that feels right.

Everything we've seen so far makes that true.

Jacob says, "I'm not sure, wouldn't it change if k were negative?" What a lovely question to pose? What do you think? Would k being negative change anything about the shape of our graph? Pause and have a think to yourself or conversation with the person next to you.

See if you can come up with an answer.

Welcome back.

Let's have a look.

Would k our constant being negative change anything about the shape of our graph? Jun says, "If we're not sure, we can test some values." So if we make k negative 10 and then we see what happens when x equals one.

Negative 10 divided by one, that's negative 10.

We plot that coordinate pair there.

When x equals two, one equals negative five, is that coordinate pair.

When x equals five, y equals negative two.

When x equals 10, y equals negative one.

Are you spotting something? How about when x equals negative one, y is positive 10? When x equals negative two, y is positive five.

When x equals negative five, y is positive two.

When x equals negative 10, y is positive one.

Oh, look.

Jun says, "I get it, when k is negative, the graph is in the opposite quadrants." Jacob agrees, "This makes sense.

A negative divided by a positive is negative and a negative divided by a negative is positive." You can always reason which quadrants you're going to find a reciprocal graph in by remembering how you calculate with negative numbers.

Quick check you've got this.

In which quadrants will the graph of y equals negative 10 over x appear? Quadrant one, two, three or four? Pause, tell a person next to you or say it aloud to me on screen.

Welcome back.

I hope you said it's not quadrant one.

It is quadrant two because we'll have negative x values and positive y values.

It's not quadrant three, and it is quadrant four.

Positive x values will give us negative y values.

In fact, there's your graph, y equals negative 10 over x.

Practise time now.

Question one, I'd like you to complete the table of values and plot the graph of y equals 20 over x.

Pause and do that now.

Question two, Laura plots the reciprocal graph of y equals eight over x.

Write a few sentences to explain Laura's error to her and show the correct position of the graph.

Pause and do that now.

For question three, I'd like you to complete the table and plot the graph of y equals negative four over x.

Pause and do that now.

Welcome back, feedback time.

Let's see how we did.

Hopefully, you populated the table of values like so and then plotted those coordinate pairs and drew those nice smooth curves representing the reciprocal graph, y equals 20 over x.

You might wanna pause, check your table of values, coordinate pairs and your curves match mine.

For question two, we're helping Laura understand her error and showing a correct position of the graph.

I asked you to write a few sentences.

You might have written, "The coordinate pairs in the first quadrant are correct.

A positive, in this case the constant was positive eight, divided by a positive will give a positive y value.

Quadrant one is correct.

However, the second quadrant is incorrect.

A positive divided by a negative will give a negative y value.

These coordinate pairs should be in the third quadrant, showing Laura that the graph should be located in quadrants one and three will help her to correct her error." Question three, completing the table of values should have looked like so.

Plotting the coordinate pairs and drawing the graph would look like so.

There we are, the reciprocal graph y equals negative four over x.

Pause and just check your values and your table match mine, your coordinate pairs on the graph match mine and your curves look just like mine.

Onto the second part of the lesson now.

We're going to look at intercepting the axes.

We've seen that graphs of the form y equals k over x do not intercept either axis, but this fact is not true for all reciprocal graphs.

If you remember back to what you know about linear graphs, the linear graph y equals two x intercepts both axes at the origin.

The line goes through the coordinate 0, 0.

When we plot y equals two x plus four, the y values are translated by positive four.

Look at our completed table of values.

All of a sudden when x equals negative two, y equals zero, that coordinate is translated by positive four in the y direction.

The same thing happens when x equals negative one, when x equals zero and so on.

And when we draw the graph of y equals two x plus four, we see something different.

Translating the y values by positive four changes the intercepts.

We've now got a y intercept at 0, 4 and an x intercept at -2, 0.

The reciprocal graph y equals two over x intercepts neither axis.

We can translate these y values by adding a constant to our equation.

y equals two over x plus one.

The table of values will be almost the same except each of those y values is one greater.

All of the y values are translated by positive one, so our coordinate pairs appear here.

When we draw the graph it looks like so.

That's the reciprocal graph, y equals two over x plus one.

We now have an x intercept, i.

e.

when y equals zero.

It's that moment there, -2, 0.

We have a moment when the equation is equal to zero.

This is because we can solve two over x plus one equals zero, i.

e.

when does the graph hit zero? Rearranging that equation, we get two over x equals negative one, x must equal negative two.

No surprises.

x equals negative two, y equals zero.

We still don't have a y intercept.

This is because dividing by zero remains undefined.

If you type two over zero plus one into your calculator, you'll find it still is undefined.

Quick check you've got this.

I'd like you to complete the table of values and decide if this reciprocal graph intercepts either axis.

Populate that table of values for the graph y equals two over x minus four.

Without even plotting, you should be able to spot any intercepts.

Pause and do this now.

Welcome back.

Hopefully, you populated the table like so.

The graph would look like this, but you knew immediately that there was not going to be a y intercept because x equals zero dividing by zero is undefined, there won't be a y intercept.

But you would have spotted in your table of values there is a moment when y equals zero.

That's where the graph intercepts the x-axis.

It happens when x equals 1/2.

Therefore, the graph intercepts the x-axis at 1/2, 0.

Just checking you've got everything we've seen today.

Reciprocal graphs intercept the axes of a Cartesian coordinate grid.

Is that true always, sometimes, or never? Pause and have a think.

Welcome back.

I hope you said that's sometimes true.

Reciprocal graphs of the form y equals k over x intercept neither axis, but reciprocal graphs of the form y equals k over x plus b intercept the x-axis provided b is not zero.

Practise time now.

I'd like you to complete the table of values and plot the graph identifying any intercepts.

That's the graph of the equation y equals eight over x plus two.

Pause and do that now.

For question two, for each equation show if the graph does or does not intercept the x-axis.

Pause and do that now.

Welcome back, let's see how we got on.

Our table of values for question one should look like so.

The coordinate pairs and the curves should look like so, and then we should identify that y equals zero when x equals negative four, therefore, the graph intercept the x-axis at -4, 0.

Pause and just check your values, your coordinate pairs, your graph look just like mine.

For question two, we didn't have to draw any graphs here.

We were just asked to show if the graph does or does not intercept the x-axis.

So will this graph ever hit zero? Essentially, solve negative 120 over x plus 15 equals zero, or can we solve it? In this case we can.

A rearrangement will tell you that 15 equals 120 over x, therefore, 15 x equals 120, x equals eight.

The graph intercepts the x-axis at 8, 0.

For b, we'd have that equation.

Will 120 over x ever equal zero? Absolutely not.

If we multiply both sides of that equation by x, we'd have 120 is equal to zero, it's not.

There's no solution.

Therefore, we know that graph does not intercept the x-axis.

For c, can we solve 15 over x minus 120 equals zero? Absolutely we can with a few rearrangements, this time x equals 15 over 120, which is 1/8.

The graph intercepts the x-axis at 1/8th, 0.

That's the end of the lesson now, sadly.

We've done a lot of learning.

We've learned that we can generate coordinate pairs for a reciprocal graph from its equation and then draw the graph.

We know that a reciprocal graph of the form y equals k over x will intercept neither axis and appear in only two quadrants.

We know that a reciprocal graph in the form y equals k over x plus b will intercept the x-axis but not the y-axis.

I hope you've enjoyed this lesson as much as I have, and I look forward to seeing you again soon for more maths.

Goodbye for now.