Lesson video

Hello.

Mr. Robson here.

Welcome to Maths.

What a lovely place it is! Today we're looking at the key features of reciprocal graphs.

Well, that sounds interesting.

Should we take a look? Our learning outcome is I'll be able to identify the key features of a reciprocal graph.

Keyword that we'll come across today, asymptote.

An asymptote is a line that a curve approaches but never touches.

For example, for the graph y equals one over x, the x and y axes are both asymptotes.

Asymptote, it's a beautiful word.

Look out for it throughout today's lesson.

Two parts to our learning today.

We're gonna start by looking at some key features.

Reciprocal graphs of the form y equals k over x will only appear in two quadrants and will not intercept either axis.

For example, this graph, y equals 10 over x, will only appear in two quadrants.

Well, why is that? Because in this case we've got a positive k value.

When we divide it by a positive x value, we'll get a positive y value.

Positive x, positive y, that coordinate pair will always appear in quadrant I.

How about when we have a negative x value? Well, we've got a positive k value in 10 divided by negative x value, we're going to get a negative y value.

Negative x negative y, that's quadrant III.

It's not possible to get a point in the other two quadrants when k is positive.

It's also true that the graphs in the form y equal k over x will not intercept either axis.

Why is that? Well, dividing by zero is undefined.

So when we try to substitute in x equals zero, 10 divided by zero, ask your calculator, that is undefined.

Therefore, a curve can't intercept the y-axis.

No matter what value x takes, there's no value that's gonna make y equal to zero.

Therefore, the curve can't intercept the x-axis either.

As the absolute values of x get really large as I can demonstrate in these table of values, I'll run x all the way to a million and I'll get a fraction one over 100,000.

I'll run x all the way to negative a million and we'll get the fraction negative one over 100,000.

The fractions will get smaller but will never reach zero.

The curve approaches but never touches the x-axis.

So we call the x-axis an asymptote to this curve.

How about as the absolute values of x get really small? If I take x all the way down to being as small as x equals one over 100,000, we get a y value of a million.

Imagine plotting that coordinate pair.

How about negative x values? When x is negative 1 over 100,000, y is negative a million.

Imagine plotting that coordinate pair.

But I hope you can appreciate when you imagine plotting them the fractions will get smaller and smaller, but will never reach zero.

The curve approaches but never touches the y-axis.

So the y-axis is also an asymptote to this curve.

For graphs in the form y equals k over x, if k is negative, some features are maintained, some are changed.

For example, y equals negative 10 over x, which feature has remained the same? Well done.

You said the x and y axes are asymptotes.

That was true of y equals 10 over x.

It's also true of y equals negative 10 over x.

So the asymptotes remained the same, but which feature has changed? Well done, the quadrants in which we see the curve.

We see this curve in quadrants II and IV, whereas we saw y equals 10 over x in quadrants I and III.

This is because we've now got a negative k value being divided by a positive x value giving us a negative y.

Positive x, negative y, that's quadrant IV.

And a negative k divided by a negative x gives us a positive y.

Negative x positive y, that's quadrant II.

It's not possible to get a point in the other two quadrants in this form.

Quick check You've got that.

Which statements are true for graphs in the form y equals k over x? Four statements there, some are true some are not.

I'll leave you to decide.

Pause now.

Welcome back.

I hope you said A is not true.

They're not linear graphs.

I hope you said B and C are both false.

The x and y axes are asymptotes to this curve.

There will be no intersections with those axes.

I hope you said D is true.

They only appear in two quadrants at any one time.

Whether k is positive, whether k is negative, they only appear in two quadrants.

Not all reciprocal graphs have the axes as asymptotes.

The reciprocal graph y equals 2 over x intercepts neither axis.

Dividing by 0 is undefined, so we have no y-intercept.

And there's no x value that will ever make y equals 0, so we have no x-intercept either.

But something's changed for the graph of y equals 2 over x plus 1.

Can you see what is different in this table of values? Pause and see if you can spot something.

Welcome back.

Did you spot that? There's an x value that makes y equal to 0, so this graph will intercept the x-axis.

When we plot the graph, there you go, negative 2, 0 intercepts the x-axis.

The y-axis remains in asymptote.

x can't equal zero because when we go to input x equals zero dividing by zero remains undefined so we don't get a y value.

The y-axis remains an asymptote.

So does this graph have a second asymptote, or does it just have one vertical asymptote, the y-xis? One way to think about it is to make the absolute x values increasingly large and see if they approach but never reach a given value.

If x equals 100, y equals one and two hundredths.

x equals 1000, y equals one and 2000ths, x equals a million, y equals one and 2000000th.

Are you seeing those y values approach but never reach a certain value? The fractional part of these mixed numbers keeps getting smaller and smaller, but the mixed number never reaches one.

How about when x values are negative? When x equals negative 100, y equals 98 over 100, negative 1000, 998 over 1000 and when x equals negative 1000000, y equals 999,998 millions.

The fraction approaches but never reaches one.

The line y equals one is now the horizontal asymptote.

And notice we changed the form of the reciprocal graph we still got a vertical and a horizontal asymptote.

Quick check you've got this.

Which of these lines are asymptotes to the graph of y equals 9 x plus 4? Four lines to choose from.

Which are the asymptotes? Pause now and decide.

Welcome back.

I hope you said the x-axis is not an asymptote.

I can see an x-intercept.

There's no y-intercept though.

The y-axis is absolutely an asymptote to this graph.

y equals 9 is not an asymptote.

y equals 4 is an asymptote.

y equals 2 over x has an asymptote at y equals 0.

We saw that y equals 2 over x plus 1 has an asymptote at y equals 1.

And we just saw y equals 9 over x plus 4 has an asymptote at y equals 4.

Have you spotted a pattern? Pause, tell the person next to you or say it aloud to me on screen.

Welcome back.

Did you notice in this form it is the constant which defines a horizontal asymptote? When we added a constant of 4 to the equation, horizontal asymptote became y equals 4.

When we added a constant of 1, we had a horizontal asymptote, y equals 1.

And when we added no constant, we had an asymptote at y equals nothing or y equals 0.

This is because y cannot take these values.

If we try to substitute in y equals 0 into that equation, it doesn't work.

2 over x can't equal 0.

The same thing for the next one.

If we substitute in y equals 1, we need 1 to be equal to 2 over x plus 1.

Well, that means 2 over x is gonna have to be 0, and that can't happen.

The same thing for y equals 4.

9 over x would have to be 0 in that case, and that can't happen.

These equations have no real solutions.

When y equals k over x plus b, the equation b equals k over x plus b has no solutions.

Therefore y equals b is an asymptote.

In the form y equals k over x plus b, y equals b is an asymptote.

If I said it twice, that probably means it's important and you might wanna pause and write that down.

Quick check you've got that.

When in the form y equals k over x plus b, which of the below are asymptotes? Four to choose from.

Pause and take your pick.

Welcome back.

Hope you said the y-axis will be an asymptote, the x-axis will not.

y equals b will be an asymptote, y equals k will not.

We can draw a visual representation of this.

In this case, the b constant is positive, but you can see the y-axis is an asymptote and y equals b is also an asymptote.

Practise time now.

For question one, I'd like you to match the graphs to the equations.

Once you've matched them up, in each case I'd like you to write a sentence to justify your decision.

Pause and do that now.

For question two, I'd like you to describe the graph of y equals 2.

4 over x minus 0.

6.

For part A, write at least three sentences and be sure to highlight key features like asymptotes and intercepts.

For part B, I'd like you to use your description to sketch the graph.

Not plop the graph, sketch it.

A sketch does not need a scale, just the axes and the line or curve.

Key features should be marked by key features.

Asymptotes, y-intercepts, mark those on your sketch.

Pause and do that now.

Feedback time.

Let's see how we got on.

Hopefully you matched these graphs like so.

Next, let's look at the sentences you might have written to justify your decisions.

In this case, A, y equals negative 9 over x, would have x and y axes as asymptotes.

You should have made that observation.

Also, the k-value is below zero.

It's negative, k is negative 9.

Therefore the curve is present in the second and fourth quadrants.

That's how you knew that graph was that equation.

For B, y equal negative 2 over x minus 9, you notice an asymptote at the y-axis.

and horizontal asymptote below the x-axis, meaning b, the constant was negative.

In this case the constant of negative 9, is how you knew it was that graph.

Additionally, k was below 0 or k was negative.

Therefore the curve is present in the upper left and lower right regions created by the two asymptotes.

For C and D.

Well, C has an asymptote at the y-axis and a horizontal asymptote above the x-axis, meaning the constant, b in this case, must be greater than 0.

By contrast, D, had an asymptote to the y-axis, but the horizontal asymptote was below the x-axis, meaning the constant there had to be negative.

In both cases k was positive, therefore the curve is present in the lower left and upper right regions created by the two asymptotes.

For question two, we were writing some sentences to describe this graph.

You might have written: this is a reciprocal graph so will form a smooth curve in two separate regions.

It will have two asymptotes.

One is the y-axis, the other is the line y equals negative 0.

6.

You absolutely should have commented on the asymptotes.

Because the y-axis is an asymptote, there is no y-intercept.

When we're describing many graphs, the y-incept is a key feature, so acknowledging that this graph will not have a y-intercept is important.

k is greater than 0, therefore the curve is present in the lower left and upper right regions created by the two asymptotes.

Next, we could look at when y equals 0.

We can solve this equation to find the intercept with the x-axis.

We can rearrange, multiply through by x, divide by 0.

6 and we get x equals 4.

The graph intercepts the x-axis at 4, 0.

When describing a graph, we're looking to describe intercepts.

In this case, we could find the x-intercept, the coordinate pair, 4, 0.

Part B, we were using that description to sketch the graph.

Our sketch should have looked something like this.

But that's not enough.

We need to label some key features so we'll acknowledge that the y-axis is an asymptote.

And we'll draw the asymptote y-equals negative 0.

6.

One last feature, that's x-intercept at 4, 0.

You want to label your sketch with those key features.

Onto the second part of the lesson now, where we're gonna be exploring asymptotes using graphing technology.

To further explore asymptotes, you'll need access to Desmos.

Open up a web browser and go to desmos.

com.

Find and press the graphing calculator button and your screen should look like this.

Pause and do all that now.

Desmos is pretty awesome.

If we type in an equation, the graph is instantly drawn for us.

Type in y equals 1 over x and you'll see that graph.

Type this one in as well, y equals 4 over x.

Pause now and make your screen look just like mine.

Welcome back.

Two reciprocal graphs in the form y equals k over x.

Key features: the x and y axes are asymptotes.

Each graph appears in only two quadrants.

Is there any value of k that changes these two key features? Now, we could alter k manually and produce several graphs.

Alternatively, we can use a feature called slider.

If you type this form in, y equals k over x, you'll be invited to add a slider, k.

Click k and add that slider.

Pause and do that now.

Your screen should look like this, in which case, you are ready to hit play and enjoy.

Do that now.

Welcome back.

Hope you enjoyed watching that.

What can we do next? Well, if you click here on the values either end of your slider, you'll see this.

From here, you can adjust the range of values for k and also this step by which it changes.

I've gone for a lower bound of negative 100 and upper bound of 100 in steps of five.

You don't have to.

You can make those values whatever you like.

One thing you might wanna do is zoom out to observe some of these changes.

Pause and have a play with that now.

Welcome back.

Was there any value of k that changed those two features? I hope you concluded, absolutely not.

There's no value of k which alters either of these two features.

The x and y axes were always asymptotes, and each graph appears in only two quadrants at any one time.

There was one exception.

The only contradiction you saw was when k equaled 0, if your slider hit 0.

In this form, when k equals 0, we no longer have a reciprocal graph, we have the linear graph, y equals 0.

For that reason, that's why you see the notation, k cannot be equal to zero.

When we communicate that a reciprocal graph is in the form y equals k over x.

Reciprocal graphs in the form y equals k over x plus b, like this one, y equals 100 over x plus 20, y-axis is an asymptote.

The horizontal asymptote is it y equals b, in this case, y equals 20 and each graph appears in only two regions.

Are there any values of k and b that change these key features? Pause, see if you can find some.

Welcome back.

No, you can make k whatever you wish, you can make b whatever you wish you'll still have the y-axis as an asymptote, you'll still have a horizontal asymptote at y equals b or y equals your constant and each graph will appear in only two regions.

So what we've seen so far is that for reciprocal graphs in both of these forms, this fact remained true: the y-axis is an asymptote.

Does that mean it will remain true for all reciprocal graphs? Luckily this is something we can explore with graphing technology.

There's another form of reciprocal graph that we've not yet met.

That's the form y equals k over bracket x plus a.

Type this equation into Desmos and see what happens.

Pause and do that now.

Welcome back.

Hopefully your screen looked like this, y equals 1 over bracket x minus 3.

This is a graph in the form y equals k over x plus a.

The y-axis is no longer an asymptote.

We have a y-intercept.

Will this be true for all reciprocal graphs of this form? Let's explore.

We could type lots of a values manually in and draw multiple graphs.

Alternatively, we can use that slider function again.

So we'll type this form into our Desmos, y equals 1 x plus a.

We'll be invited to click a to add a slider, and we absolutely want to do that.

Pause, make your screen look like mine.

Next, we're ready to hit play and observe what happens as Desmos changes the values of a.

Do that now.

Welcome back.

What did you notice? I hope you noticed.

As a value of a changes, the x-axis remained an asymptote this time, but the position of a vertical asymptote changes.

There is one moment when a equals 0 that the y-axis is an asymptote, but outside of that the graphs have a y-intercept.

True or false? Reciprocal graphs always have an asymptote at the y-axis? Is that true? Is that false? And use one of those two statements at the bottom to justify your decision.

Pause, and answer this now.

Welcome back.

I do hope you said false and used statement B to justify your decision.

Reciprocal graphs in the form y equals k over x plus a intercept the y-axis provided a is not equal to 0.

In the form y equals k over x plus b, we saw horizontal asymptotes at y equals b.

Can we make such a generalisation for the vertical asymptotes for the form y equals k over x plus a? I think you know we can, and I think you're looking forward to learning what that generalisation is.

Type these equations into Desmos and try to identify the equation of the vertical asymptotes.

You've got this.

I'm gonna leave you to do this.

Pause now.

Welcome back.

Let's see what you found.

Hopefully you noticed that the graph y equal 1 over x minus 4 had an asymptote at x equals 4.

1 over x minus 2 had an asymptote at x equals 2.

1 over x plus 3 had an asymptote at x equals 3.

The position of these asymptotes makes perfect sense.

A question we should ask ourselves is what value makes the denominator 0? Because we know dividing by 0 is undefined.

In the first case when x equaled 4, the denominator is 0, dividing by 0 is undefined.

So we've got an asymptote at x equals 4.

For the second graph y, when x equals 2, the denominator is 0 and we know that dividing by 0 is undefined, hence we get an asymptote at 2.

You know what's coming next.

For the last equation, when x equals negative 3 the denominator zero, and dividing by zero is undefined, hence we get an asymptote at x equals negative 3.

So a key question then, what x value makes this denominator 0? Well done.

It's when x equals negative a.

So graphs in the form y equals k over x plus a have vertical asymptotes at x equals negative a.

You might wanna write that down.

Pause now.

Quick check you've got that.

What is the equation of the vertical asymptotes of this graph, y equals 975 over x plus 165? Pause and take your pick.

Welcome back.

I do hope you said B: x equals negative 165.

That's what that graph would look like.

And you knew that would be the vertical asymptote because negative 165 is the x value that makes this denominator equal to 0, therefore undefined.

Practise time now.

For question one, I'd like to draw a reciprocal graph that does not intercept the x-axis or intersect the line, x equals negative 5.

You might wanna use graphing technology to explore this question.

Pause and do it now.

For question two, I'd like you to draw a reciprocal graph that does not intercept the y-axis or intersect the line, y equals negative 3.

Again, experimenting with our graphing technology is gonna help you find the answer to this one.

Pause and do that now.

Question three, tricky one, but a lovely one.

You'll enjoy this.

Fill in the blanks to draw the reciprocal graph that does not intersect either the lines x equals negative 4 or y equals negative 2.

And also identify where the curve intersects the axes.

There's four blanks there in that equation.

What you gonna put there? Pause, see if you can work this one out.

Feedback time now.

Let's see how we got on.

You know that this will be a graph in the form y equals k over x plus a.

The vertical asymptote that's moved, it's no longer the y-axis.

k can take any value, but a must have a value of positive 5.

So the the example I found that worked was y equals 4 over x plus 5.

You could have any value, non-zero, over x plus 5.

You could have had 40, 400, but your denominator had to read x plus 5.

For question two, this is gonna be a graph in the form y equals k over x plus b.

k can take any value, but b must have a value of negative 3.

So the example like I drew that worked, y equals negative 9 over x minus 3.

You could have had negative 900 over x minus 3, but you had to have minus 3 in the b position.

For question three, filling in the blanks to draw a reciprocal graph that intersects neither of those lines.

Tricky one this, but a lovely problem.

Hopefully, you had a denominator of x plus 4 and added a constant of negative 2.

That gives us that graph, y equals 20 over x plus 4 minus 2 giving us two intercepts at 0, 3 and 6, 0.

Sadly, right at the end of the lesson now.

But we've learned that we can identify the key features of a reciprocal graph.

We know that sometimes, but not always, the x or y-axis may be asymptotes, but that there are forms of reciprocal graphs that intercept these axes.

Hope you've enjoyed this lesson as much as I have.

And I'll look forward to seeing you again soon for more maths.

Goodbye for now.