Lesson video

Hello, Mr. Robson here.

Welcome to maths.

What a lovely place it is.

Today we're looking at the key features of cubic graphs and cubic graphs are awesome, so this lesson should be awesome.

Let's take a look.

Our learning outcome, is that we'll be able to identify the key features of a cubic graph.

Some keywords we're gonna come across today, cubic.

A cubic is an equation graph or sequence whereby the highest exponent of the variable is 3.

The general form for cubic, is axe cubed + bx squared + cx + d.

Roots.

That's another word you're gonna hear a lot.

When drawing the graph of an equation, the roots of the equation are where its graph intercepts the x-axis where y equals zero.

For example, this quadratic has roots at x = 2 and x = 5.

Turning point.

The turning point of a graph, is a point on the curve where as x increases, the y values change from decreasing to increasing, or vice versa.

We'll have two parts to our learning today and we're gonna begin by finding the roots.

Before we look at cubics, let's start with something we're very familiar with.

Quadratic equations.

Quadratic equations can have a varied number of solutions.

The equation -x squared + 6x - 7 = -2 has two solutions.

That's there.

Why? Because there's two intersections between those two lines, therefore, there's two solutions to that equation.

The equation -x squared + 6x - 7 = 2 has one solution.

How so? Well, when we draw the line, y=2, the lines have one intersection, therefore there's one solution.

The equation -x squared + 6x - 7 = 3 has no solutions.

Why is that? Well, when we draw y = 3, there are no intersections between the parabola and the horizontal line.

Therefore, there's no solutions to that equation.

Let's explore the solutions of cubic equations.

The equation x cubed - 2 = 3 has one solution.

It's there.

One intersection, therefore one solution.

The equation x cubed - 2 = -1 has one solution.

There it is.

We've got one intersection, therefore one solution.

How about x cubed - 2 = -2, that's got one solution as well.

There it is.

We have one intersection, we have one solution.

Does this mean that all cubic equations have one intersection and one solution? What do you think? What's your mathematical intuition telling you? If we plot the graph of y = x cubed - 3x + 3, we notice something that contradicts that.

The notion that cubic equations have one solution.

Let's populate the table of values and we notice something.

Repeated y values in the table, show that there must be moments with more than one intersection, more than one solution.

Look at those y values.

We see 1 twice, we see 5 twice, so that means there must be more than one intersection at some point.

When we draw that cubic curve, it looks like so.

We start to look at some equations.

The equation x cubed - 3x + 3 = 6 has one solution.

There we are.

One intersection, one solution.

But the equation x cubed - 3x + 3 = 5 has two solutions.

When we draw the horizontal line, y = 5, oh look, two intersections, therefore two solutions.

The equation x cubed - 3x + 3 = 1 also has two solutions.

There's the line y = 1 and we can see two intersections.

Therefore two solutions.

The equation x cubed - 3x + 3 = 3 has three solutions.

Draw the line of y = 3 and look at that, three intersections, therefore, three solutions.

So in answer to our question, does this mean that all cubic equations have one intersection, one solution? The answer is categorically no.

It is possible for cubic equations to have one, two, or three solutions.

Quick check you've got that.

This is the graph of y = -x cubed + 6x squared - 9x - 2.

I could use that graph, to show you that -x cubed + 6x squared - 9x - 2 = -8 has one solution.

What I'd like you to do, is insert integer values into those three boxes, to make the three statements true.

Give me an integer value that gives us one solution, an integer value that gives two solutions an integer value that gives us three solutions.

Pause and do that now.

Welcome back.

Let's see how we did.

Hopefully, for the first one, you put something like -7.

If I use the example of -7, we can see that that has one intersection, one solution.

You don't have to use -7 though, any value of equal to or less than -7, or greater than or equal to -1, would make this statement true.

An equation that has two solutions, you could have said -6.

You weren't limited to -6.

If you'd said -2, that would also have been true, but only -6 and -2 make this statement true.

How about three solutions? Well, I'm gonna use -3.

Oh look, three intersections, three solutions.

You weren't limited to saying -3 for that last one.

You could have said -3, -4 or -5.

Any of those would've made that last statement true.

The graphs of the cubic equations we've seen so far have all had one root.

Just to remind you, a root is where the graph intersects the x-axis.

There's the roots, one root for each of those equations.

Does this mean that all cubic equations have one root? What do you think? What's your mathematical intuition telling you? Hopefully, it's telling you no.

If cubic equations can have one, two, or three solutions, then cubic graphs can have one, two, or three roots.

If we look at these three examples, our first example has one root.

Our second example has three roots and the third example has two roots.

Notice, that when we have two roots, there's something special about one of the roots.

If a cubic graph has two roots, one of them must be a repeated root.

For this graph, x = 4, is a repeated root.

Quick check you've got this.

A true or false.

Cubic graphs have only one root.

Is that true or is it false? And select one of the two statements at the bottom to justify your answer.

Pause and do that now.

Welcome back.

I hope you said false and I hope you justified that with the statement, "Cubic graphs can have one, two, or three roots." Well done.

Before going on to understand something more about the roots of a cubic, I'd like to go back to quadratic graphs and remind you that the quadratic graph y = x squared + 8x + 15, has two roots.

It's got root set x = -5 and x = -3.

The roots are when the quadratic is equal to zero.

We could factorise this, into x + 5, x + 3.

Hence, we see the roots at x = -5 and x = -3 They're the only two x values, that'll make the equation equal to zero.

We could write an equation with roots x = 2 and x = 4.

This is working backwards, if you will.

If I want roots at x = 2 and x = 4, I'm gonna make the brackets x - 2 and X - 4.

If I expand those brackets, I get the quadratic equation x squared - 6x + 8.

When I look at the graph of that quadratic, Oh look, roots at x = 2 and x = 4.

We started with our roots and created an equation.

So what happens when we try to write an equation with roots at x = 1, x = 2 and x = 4? We'd need those brackets, x - 1, x - 2 and x - 4 because when I substitute in any of those three x values, y will be equal to zero, hence x = 1, x = 2 and x = 4 will be our roots.

I'm gonna expand these brackets now, but I won't do all three at once.

I'll expand the first pair and get x squared - 3x + 2.

Then I'm gonna multiply those two brackets together and I will get the expression x cubed - 7x squared + 14x - 8.

When I graph it, oh look, a cubic graph, of course.

The highest exponent of x is 3.

And look, the roots at x = 1, x = 2 and x = 4.

How nice.

This approach explains where a repeated root comes from.

Let's look at this quadratic, y = bracket x - 3 squared.

If we expand that, we get x squared - 6x + 9 and when we graph it, it looks like so and there we are.

A quadratic with one repeated root, at x = 3.

If we look at this equation instead, not just x - 3 squared but that multiplied by x + 1.

Let's expand the first bracket and then we'll expand this pair and we get y = x cubed - 5x squared + 3x + 9.

When we graph that, it looks like so and no surprises.

We've got a cubic, with one root, x = -1 and one repeated root, at x = 3.

Let's check you've got that.

I'd like you to identify the roots, to match the graphs to their equations.

As three graphs, what are their roots? Therefore, which equation is which graph? Pause and do this now.

I hope you matched equation A to the third graph because there were three distinct roots, one of them, positive.

I hope you matched equation B to the middle graph because there were three distinct roots and all three roots were negative.

Finally, we'd match C to the first graph because it had two roots one of which is a repeated root.

You can see the repeated root in the equation, where it says x + 5 squared.

You can see the repeated root in the graph at x = -5.

Not all roots will be integer values and not all cubics will easily factorise.

In these cases, we might be asked to estimate the roots of an equation.

For this equation, I can pick out the roots and estimate them to one decimal place.

My estimated roots are x = -1.

4, x = -0.

7 and x = 0.

9.

Your turn.

I'd like you to estimate the roots of this cubic equation.

One decimal place will do it.

Thank you very much.

Pause.

Estimate those roots now.

Welcome back.

Let's see how we did.

Hopefully you picked out those roots and estimated them to be x = -0.

2, x = 0.

8 and x = 2.

2, to one decimal place.

Well done.

Practise time now.

Question one, I'd like you to find the roots of the graphs below.

Pause and do that, now.

Question two, for the cubic equation y = x cubed - x squared - 5x + 1 I'd like you to complete the table of values.

Then plot the coordinate pairs, draw the graph of the equation and finally for part C, estimate the roots.

In your estimations, one decimal place will do.

Pause and do this, now.

Welcome back.

Let's see how we did.

Find the roots using the graphs below.

For A, you should have found a root at x = 2.

For B, we should have found a root at x = 4 and a root at x = 7.

You should have noted, that the root at x = 4 is a repeated root and for C we had roots at x = -2, x = 1 and x = 4.

For question two, completing the table of values would give you those y values.

Plotting the graph, it looks something like that.

Picking out the roots and then estimating those roots to one decimal place.

We should get x = -1.

9 x = 0.

2 and x = 2.

7, to one decimal place.

You might wanna pause, just check your table values matches mine, your graph matches mine and your roots are accurate to one decimal place.

Onto the second half of the lesson now, where we're going to look at turning points.

Again, let's think about quadratics before we think about cubics.

Quadratic graphs, form parabolas, which have turning points.

The turning point of a graph, is a point on the curve where as x increases, the y values change from decreasing to increasing or vice versa.

For this example, the y values change from increasing to decreasing, so we call this turning point a maximum.

The turning point is at 4, 8.

There will be no higher y value than 8 anywhere on this graph.

That is why that turning point is a maximum.

Cubic graphs form cubic curves which also have turning points.

At this point, the y values change from increasing to decreasing.

Hence, it's a turning point.

What is different about this turning point versus the one we saw on the parabolic graph? Pause.

Have a conversation with the person next to you or a good think to yourself, see if you can answer that question.

Welcome back.

I wonder what you thought.

Hopefully, you said something along the lines of this is not the maximum point on this graph.

There will be infinitely more y values higher than this one.

Whilst it may not be the maximum point on this graph, we need an efficient way to communicate, that it's a turning point, at which the y values change from increasing to decreasing.

If we focus just on this interval around this point, it's the maximum point in this locality.

Just in between those two vertical lines, in that short interval, is the maximum point in there.

If it's the maximum point in that locality, would it surprise you to learn that's why we call it a local maximum? We can label that as local maximum.

Then, if we call this point a local maximum, what do you think we call this point? I do hope you said local minimum.

We call it a local minimum.

We can label it local minimum.

Why is it a local minimum? Because in this particular interval, this is the minimum value the graph reaches.

Quick check you've got this.

Which of these statements are true? There's three statements there.

Read them all.

Pick out which ones are true, which ones are false.

Pause and do this, now.

Welcome back.

Let's see how we got on.

Hopefully, you said A was false.

There will be infinitely more higher y values on the graph.

I hope you said B is true.

On cubic graphs we need local maximums because they tell us the highest y value, in that locality.

And I hope you said C was false.

A local maximum tells us that y values have gone from decreasing to increasing? No.

A local maximum tells us that y values have gone from increasing to decreasing and that's important.

Next, I'd like you to apply the labels to this cubic graph.

If you can apply those labels, you've picked out a lot of the key features of this cubic.

Pause and do this now.

Welcome back.

Let's see how we did.

Hopefully, we declared that to be a root, and that to be a local minimum.

The y-intercept, another root, local maximum and a root.

There we are.

The key features of the cubic.

Next, I'd like you to complete the statements.

The cubic equation y = 2x cubed + 21x squared + 60x + 52 looks like so.

There's four blank spaces on those statements.

I'd like you to fill them, but with what? I'll leave you to decide.

Pause and have a think.

See you in a moment.

Welcome back.

Let's see how we did.

The cubic equation has a root at x = -6.

5.

We should have written a root there, x = -6.

5.

It has a local maximum at -5, 27.

We should have written a coordinate there.

Has a repeated root at, well done x = -2.

We write a root in that space.

This repeated root is also a.

well done.

It's a local minimum.

We needed a keyword there.

Local minimum.

Practise time now.

Question one, here's a graph for a cubic equation.

For part A, I'd like you to identify the coordinates of the turning points.

For part B, I'd like you to label each turning point, local maximum and local minimum and write at least two sentences to justify why the points get those labels.

Pause and do this, now.

Question two, and this question is lovely.

It's gonna be an enjoyable mathematical wrestle, but we'll get a lot outta it.

Write a description of the graph of the equation.

y = x cubed - 6x squared + 9x, identifying the y-intercept, the roots and turning points.

We don't need to plot that graph to write a description.

If we're not going to plot it, then you will need to know: Firstly, the equation factor rises simply once you take out the factor x.

Secondly, you'll want to know there's a turning point when x = 1.

Finally, you'll benefit from knowing that one turning point is a repeated root.

Pause.

Have a good play with the maths here and write a description of that equation.

Welcome back.

Let's see how we did.

Question one, part A, I asked you to identify the coordinates of the turning points and we should have identified them as -1, 7 and 2, -20.

They are our turning points.

Secondly, I said label them as local maximum and local minimum.

Local maximum there.

Local minimum there.

And finally, I asked you to write at least two sentences to justify why the points get those labels.

You might have written -1, 7 is a local maximum because it's a turning point where y values go from increasing to decreasing.

Whilst not the maximum point on this graph, it is the maximum point in this locality.

If you didn't get all that detail down in your sentence, maybe pause and copy it down now.

As for our local minimum, 2, -20 is a local minimum because it's a turning point where y values go from decreasing to increasing.

Whilst not the minimum point on this graph it's the minimum point in this locality.

Again, pause.

Make sure you have all that detail in your answer.

Let's see how we did on question two.

This question was delightful.

We need to remember the equation factor rises simply once you take out a factor of x.

There's a turning point when x = 1.

One of the turning points is a repeated root.

Okay, we're ready.

Let's start by taking out the factor of x and then did we notice something? Did you notice x squared - 6x + 9 is a perfect square? We've turned our equation into y = x, multiplied by x - 3 squared.

It must have roots at x = 0 and x = 3.

They're the only two values which will make y equal 0.

Hence, those x values are our roots.

When x equals 0, y equals 0.

There's our y-intercept.

When x = 1, y = 4.

There's a turning point at 1, 4.

Remember we were told there was a turning point when x = 1.

We know that turning point is a local maximum 'cause y values change from increasing to decreasing.

How do we know that happened? Well, when x was 0, y was 0.

Then when x was 1, y was 4.

Ah.

Y had increased.

If they're increasing before the coordinate, 1, 4 they must be decreasing after.

That must be a local maximum.

3, 0 must be a turning point.

We were told the repeated root was a turning point.

It is a local minimum because the other turning point was a local maximum.

We put all that together.

A description might read, the graph formed a cubic curve with root sets x = 0 and x = 3.

X = 3 is a repeated root.

It has a turning point at 1, 4, which is a local maximum.

It has a turning point at 3, 0, which is a local minimum.

It has a y-intercept at 0, 0.

Now it's useful, when you're doing a question as complex as this to use tech, to check.

I think we're right, but if we type that equation into graphing technology such as Desmos, we'll see the graph.

The good thing about that graphing technology, it picks out these key features for us.

The y-intercept 0, 0, the first turning point at 1, 4.

The repeated root and local minimum at 3, 0.

That was really useful, using tech to check.

Sadly, at the end of the lesson now.

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

A cubic graph can have one, two or three roots.

It might have turning points which are not maximum, minimum values of the whole graph, so they are called local maximums and local minimums. Hope you've enjoyed this lesson as much as I have and I look forward to seeing you again soon for more mathematics.

Goodbye for now.