Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Today we're drawing cubic graphs and they are gorgeous.

So let's get started.

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

Keywords, we'll see the word cubic.

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

The general form for a cubic is AXE cubed plus BX squared plus CX plus D.

You'll hear that word a lot today.

Cubic.

Two parts to our lesson.

We're gonna begin by plotting cubics.

Let's start with a little look at language.

When graphing equations, we call Y equals X a linear graph because when plotted it forms a straight line.

We call Y equals X squared a quadratic after quadratum, the Latin word for square, the highest exponent is two, X is being squared.

It therefore makes sense that we call Y equals X cubed a cubic because the highest exponent is three and X is being cubed.

We can plot Y equals X cubed with a table of values.

I'm gonna start by substituting in the most simple X values.

X equals zero.

When X equals zero, Y equals zero.

When X equals one, Y equals one.

When X equals two Y equals eight, I'll pop those into my table.

I'm now gonna tackle the two trickier X values when X equals negative one and when X equals negative two.

This is where people frequently go wrong when populating a table of values.

So let's take a closer look at the right thing to do.

Y equals negative one cubed.

That means negative one repeatedly multiplied by itself three times negative one multiplied by negative one multiplied by negative one.

That makes negative one.

Why? Because negative one multiplied by negative one is positive one, but when we multiply it by negative one again it becomes negative one.

Similar thing happens for negative two.

Negative two multiplied by negative two is positive four.

But when we multiply it by negative two again we get to negative eight.

It's really important that you remember that when you cube a negative value, the result is negative.

So when X equals negative one, Y equals negative one, when X equals negative two, Y cause negative eight.

Pop those into our table of values and we're ready to plot those coordinate pairs.

So what do we do from here? Do we join these points with line segments? Should this graph look like that? If this were the case, we would see the coordinate half half, but we won't see that.

In fact, we'll use that coordinate to justify or the answer to that question is no.

We can justify this because the values and coordinate pair a half a half do not satisfy the equation.

When I substitute a half into the Y position in the equation and a half into the X position in the equation, the equation is no longer satisfied because a half cubed is not equal to a half.

If we substitute correctly when X equals a half, Y equals a half cubed, which is one eighth.

If we plot this graph in steps of a half, we start to see its shape.

There's the graph in steps of a half.

What if we did it in increments of a quarter? Are you starting to get a picture of what this graph looks like? If we plot this graph in infinitesimally small steps, we'll see that it forms a smooth curve.

We call this a cubic curve.

Quick check you've got this so far, which of the below statements are true.

There's three statements there I'd like you to read each of them and consider which ones are true.

Pause and do that now.

Welcome back.

Let's go through them one by one.

A is not true.

When plotting cubic graphs, we join our coordinates with straight line segments.

Absolutely not.

Cubic graphs form a smooth curve.

Once we plotted our coordinate pairs, we want to draw a smooth curve through them.

B is also not true.

Cubing a negative will not produce a positive result.

Cubing a negative produces a negative.

For example, negative five cubed is negative 125.

C was also not true.

Quadratic graphs form a parabola.

Cubic graphs form a cubic curve.

So what's the same and what's different? When we plot Y equals X cubed plus five, some things changed about that equation.

Will they change our graph? How will it change our graph? I don't know.

So let's explore.

We're gonna populate a table of values.

I'm gonna start with the simplest X values.

When X equals zero, Y equals five.

When X equals one, Y equals six.

When X equals two Y equals 13, I'm quite confident that they're correct, but I'm gonna take time and pay particular attention when I'm substituting in X equals negative one and X equals negative two because this is where people typically go wrong.

When X equals negative one, Y is going to be negative one cubed plus five.

That'll be negative one plus five.

So positive four.

When X equals negative two, negative two cubed plus five becomes negative eight plus five, which is negative three.

Take time, care and attention in those moments.

We'll pop those into our table of values and we'll plot those coordinate pairs.

When we draw a nice smooth curve, we notice a few things.

What's the same? What's different? Same.

It's the same shape.

It's a cubic curve again, what's different? The position of the curve.

When we added a constant positive five to the equation, we changed the Y intercept to zero five.

Well that makes sense because when X equals zero Y equals five, no wonder we've got a Y intercept at zero five.

Quick check you've got that.

Laura rushed when completing this table of values for Y equals X cubed minus two.

Laura says, I have a horrible feeling I've made some errors.

Could you use your calculator and check my work please? That's a super idea.

I'd like you to have a go at that now can you use your calculator substitute in those X values and check Laura's Y values? Are they right? Are they wrong? Which ones are right? Which ones are wrong? That's down to you to figure out.

Pause and do that now.

Welcome back, let's see how we did.

Hopefully when you substituted in X equals negative two into the equation, your calculated display looked a little like that and you were able to say that's not right, when X equals negative two, Y should be equal to negative 10.

From here we can use our left arrow key to go back, change that negative two to a negative one and we've got the Y value when X equals negative one, it should be negative three, not negative one.

When X equals zero, Y is indeed negative two.

Laura is right there when X equals one, Y is indeed negative one.

Laura is right there, but when X equals two, Y should equal six.

That one was not correct.

Laura says, I see what I did wrong.

I thought negative one cubed was positive one, and for two to the power of three I multiplied by three instead of cubing.

These are common errors.

What Laura's done really well there is she's reflected on her errors and she's learned what she won't do next time.

Well done Laura.

When we plot the coordinate pairs and we draw a nice smooth curve, it's no surprise that we see a cubic curve, but this time with a Y intercepts at zero negative two.

What effect will subtracting the cubed X term have? If we plot Y equals five minus X cubed, I wonder, let's explore again, no change.

Let's start with the easiest X value when X equals zero five minus zero cubed, is just five five minus one cubed is four, five minus two cubed is minus three or should I say negative three.

Let's pop those in our table and then let's take a lot of time, care and attention when substituting an X equals negative one.

Five minus negative one cubed, well that'll become five minus negative one, which is five plus six.

Five minus negative two cubed.

That's gonna be five minus negative eight becoming positive 13.

Pop those into our table of values.

We plot those coordinate pairs and draw a nice smooth curve through them.

We start to see some key features.

We got Y intercept at zero five.

It's no surprise in our equation we had a constant of five, so an X equals zero, Y was always going to be positive five.

We plot the same shape.

It's a cubic curve.

However, the direction of the curve has changed.

Can you see that as the values for X increase, the values for Y tend to decrease.

That's because look at that X cubed coefficient.

It's negative.

It's causing those Y values to tend to decrease as Y increases.

Quick check you've got this.

What do we know about the equation of this cubic? There's four statements there, some of them are true.

Pick out some features for that graph.

Pause and do this now.

Welcome back, I hope you picked out B.

The coefficient of execute is negative for this equation and D, the constant is negative.

This was the graph of Y equals negative seven minus X cubed.

We've got a negative X cubed coefficient.

As the X values increase, the Y values tend to decrease and we've got a negative constant giving us a Y intercept that's below the X axis.

For this graph that Y intercept was zero negative seven.

Practise time now, question one, let's complete the tables of values and draw these cubic graphs.

For A we're gonna draw Y equals X cubed minus eight, and for B we're going to draw Y equals 12 minus X cubed.

Pause and do this now.

Question two.

I'd like you to match these cubic equations to their graphs and in each case I'd like you to write a sentence to justify your answer.

Pause and do this now.

Let's see how we got on.

The table of values for the graph Y equals X cubed minus eight should look like so and when you plot it, your cubic curve should be there.

For B you should have that table of values and that cubic curve.

You'll want to pause at this moment and check that your value Y values match mine.

Your coordinate pairs match mine and your cubic curves look just like mine.

For question two, we are matching equations to their graphs.

Hopefully you matched A to the third graph, B to the second graph and C to the first graph.

In terms of sentences to justify our answers, you might have written that that graph was A, Y equals X cubed minus 32 because it's a positive X cubed coefficient and a negative constant, which gives us a Y intercept below the X axis.

A sentence to justify why that is the graph of Y equals X cubed plus 24 might be a positive X cubed coefficient and a positive constant giving us a Y intercept above the X axis.

Finally, the C, Y cause 19 minus X cubed a cubic curve with generally decreasing Y values.

Therefore a negative X cubed coefficient and a positive constant which gives us a Y intercept above the X axis.

Onto the second half of the lesson.

Now we're going to look at more complex cubics.

Something interesting can happen when we introduce more terms involving X.

For example, Y calls X cubed minus 3X squared plus 2X.

A calculator can speed up substituting a value into multiple terms in an expression.

I wanna substitute X equals negative one into that expression.

I'm gonna do it like so.

And you'll notice I've used brackets.

What this means is I can easily identify and change the value of the variable.

I know that when X equals negative one, Y equals negative six and I can very quickly use the left arrow key, change the negative one to a zero on my calculator.

Therefore, when X equals zero, Y equals zero, keep going with that.

And we find when X equals one Y equals zero when X equals two Y equals zero.

And when X equals three, Y equals six.

These seem like unusual results.

The same value zero appears three times in our table.

When we come to plot those coordinate pairs, we see this.

What do you think happens here? Zero zero, one zero, two zero.

Do we just join them with a horizontal line segment? What do you think? I'm hoping you said no, but how do we know that that's not the case? What we can do is test any value in between any of those X values.

Between zero and one is a half.

I'm gonna test what happens in our equation when X equals a half.

When X equals a half, Y will be a half cubed minus three lots of a half squared plus two lots of a half.

That's three eighths, so we'll have a coordinate a half, three eighths there we know it's not a straight line segment in that moment now.

What if we plot the whole graph in steps of a half, then we'll start to see the shape.

We can increase those increments in steps of a quarter it looks like so and we're starting to see the shape.

If we plot this in infinitesimally small steps, we find it makes a smooth curve.

This is still a cubic curve.

It's just one with multiple turning points and multiple roots.

Quick check you've got this.

Laura forgot her calculator today but still tried to complete this table of values 4Y equals X cube minus 5X squared plus 4X plus six.

Laura says, I have a horrible feeling of made some errors.

I got six three times.

Could you use your calculator and check my values please? I'd like you to do that.

Pause and check the accuracy of Laura's Y values.

Do that now.

Welcome back.

Hopefully your calculator display looked something like that when you substituted in X equals negative one.

We find that the Y value should be negative four, so that one wasn't correct.

We can go and change that X equals negative one to an X equals zero and find that the Y value was indeed six.

When X equals one, the Y value was indeed six.

When X equals two, Y does indeed equal two.

When X equaled three, the Y value was zero.

That's another error.

And when X equaled four Y equals six.

It's not unusual to see repeated values like six in this case when plotting a complex cubic.

So don't be afraid when you see it, but do always check your work.

Laura says, thank you.

I'll remember my calculator next time.

It's a super important piece of kit.

I couldn't agree more Laura.

Laura now has the correct coordinates for Y equals X cubed minus 5X squared plus 4X plus six.

So she tries to draw the graph.

She hasn't seen one as complex as this before, so she draws this curve.

Would you have drawn it like this or would you have drawn something different? Pause, have a conversation with a person next to you or a good think to yourself.

You've done that or something different.

See you in a minute.

Welcome back, hopefully you recognised this is not a cubic curve, so I absolutely wouldn't draw that curve.

When graphing You need to go through the coordinates in ascending X value.

So a line should go through negative one, negative four, zero six, and then one six and then two two and then three zero, and then four six.

This is now a cubic curve.

Laura says, yes, that graph makes way more sense.

Thank you.

You are welcome Laura.

Quick check you've got this, which statement is true when drawing cubic curves statement A, statement B, or statement C.

Have a read, take your pick, pause.

I'll see you in a moment.

Welcome back.

Hopefully you said A join the coordinates in ascending X value order.

When you do that, you get a nice smooth cubic curve like so.

Practise time now, question one.

I'd like you to complete the table of values for the cubic equation Y equals X cubed minus X squared minus two and draw the graph, pause and do that now.

Question two I'd like you to complete the table of values for the cubic equation Y equals negative X cubed plus X minus two and draw the graph, pause and do that now.

Then for question three, complete the table of values for the cubic equation Y equals X cubed plus 2X squared minus 8X minus 13 and draw the graph.

You are welcome to use your calculator for this one.

Let's see how we got on.

Your table of values for question one should look like so, and then your coordinate pairs and your cubic curve would look like that.

You might wanna pause now and just check that your coordinate pairs your table of values, your curve match mine.

For question two, table of values should look like so let's not be afraid when we see negative two, three times in our table.

When we join those coordinates with a nice smooth curve, we get a graph like that.

Again, pause, check your table of values, your coordinate pairs, and your graph match mine.

For question three, there's just a couple of missing values in the table.

We should have populated them like so.

Those we are missing Y values three negative four, negative 18, negative 13.

When plotted, you get those coordinate pairs.

Hopefully you went through them in ascending X order and got a cubic curve that looks like so.

Again, pause, check your values, your coordinate pairs, and your curve match mine.

Sadly, we're at the end of the lesson now, but we've learned that we can generate coordinate pairs for a cubic graph from its equation.

It's important to take time and check for accuracy when substituting a value into multiple terms of an expression.

We use a calculator when permitted.

The graph of a cubic equation forms a cubic curve, and I hope you agree that all the cubic curves we saw today were beautiful.

I look forward to seeing you again soon for more beautiful mathematics.

Goodbye for now.