Lesson video

Hello, Mr. Robson here.

Welcome to maths.

Today we're going to use technology to explore the shape of graphs.

Well, graphs are awesome, technology's awesome, so that should make this lesson awesome.

So let's get on with it.

Our learning outcome is I'll be able to appreciate that different types of equations give rise to different graph shapes, identifying quadratics in particular.

Some keywords that are gonna come up throughout this lesson.

The relationship between two variables is linear if when plotted on a pair of axes a straight line is formed.

Linear, that's a word I imagine you're familiar with.

Quadratic might be a new one to you.

A quadratic is an equation whereby the highest exponent of the variable is two, the graph of which forms a parabola.

For example, x squared plus x is quadratic, whereas x cubed plus x is not.

A parabola.

That word might be new to you.

A parabola is a curve where any point on the curve is an equal distance from a fixed point, which we call the focus, and a fixed straight line, which we call the directrix.

The line of symmetry goes through the focus and is at right angles to the directrix.

Two parts of today's lesson, we're gonna begin by reviewing plotting graphs of equations.

To get started, open up a web browser, go to desmos.

com.

Once you're there, find and press the graphing calculator button and your screen should look like this.

Pause this video, get caught up.

Make your screen look just like mine.

Welcome back.

Assuming you are now at desmos.

com, I'd like you to click on the graph settings menu in the top right of your screen.

It's that little spanner icon up there.

Once you click on it, it'll reveal this.

I'd like you to make some changes here.

To begin with, we're gonna turn off the minor grid lines, so unhighlight that box.

Then I'm gonna type x into the x-axis label box and y into the y-axis label box.

Funnily enough, that's going to label our axes for us.

Then I'd like you to change the steps on both axes to one.

Pause and make those changes.

Welcome back.

Your grid should look a little like this.

We're gonna need to navigate around this.

You can zoom in and out using the positive and negative buttons up in the top right of your screen there.

Alternatively, you can zoom by holding down the Shift button on your keyboard, clicking and holding the mouse button and moving that mouse back and forth.

That'll zoom in and out for us, but we won't just want to look at this central region of the graph.

We'll want to scroll all over the place.

By clicking and holding the mouse button, you can drag your axes around the screen to look at different regions.

Can you pause this video and just check that you know how to zoom in, zoom out and know how to scroll to look at different quadrants at different times.

Pause.

Do that now.

Welcome back.

We're now ready to explore plotting graphs of equations.

I'd like you to start by zooming and dragging so your screen is scaled with negative five to positive five visible on the y-axis and zero to positive five on the x-axis.

We're gonna use the table function to plot our first graph, so click on the add item button and click table.

A table should appear.

We can enter coordinate values into this table.

The first graph that we're going to plot is the linear graph of y equals 2x minus five for x values in the range of zero to positive five.

We'd like to type zero to positive five into the x column.

My hint for you is type in zero and then underneath it, one and then hit enter.

It should auto populate in steps of one for you.

Did it work? Does your x column look like mine? I hope so.

We're now ready to find the respective y coordinates for these x coordinates.

The function is y equals 2x minus five, so when x equals zero, y equals two lots of zero minus five, i.

e.

it's minus five.

That's gives us the coordinate zero, negative five.

When x equals one, y equals two lots of one minus five, it's minus three.

That will generate the coordinate one minus three.

We type those into our table of values and you can see those two coordinates are plotted for me.

We need to find the remaining y coordinates and add them to the table.

They should look like so and we should get those coordinates plotted like so.

I'd like you to click and hold here to change the dots to crosses.

Now, you will see coordinates plotted as dots.

All you mathematicians like to be as accurate as possible and crosses are a little more accurate than dots, so we should have coordinates like so.

Once we've got it like them, they form a line and we can reveal that line by and holding here and turning on the lines button.

Once you've done that, your graph should look just like this.

That's the linear graph of y equals 2x minus five for the given range zero is less than or equal to x is less than or equal to five.

Can you pause this video and make sure that you can make your table of values, your coordinates and your line look just like this? Welcome back.

Let's take a closer look at that table of values for it will reveal something about this graph.

Look here in the y column.

For every change of positive one in x, what do you notice about y? Well done, a change of positive two, a common constant difference, hence we get a linear graph.

Quick check you've got that now.

I'd like to plot the linear graph of y equals four minus x for x values in the range of zero to positive four.

I've demonstrated what the table will look like in your Desmos and I've given you the first couple of lines to working out the respective y coordinates.

Can you plot that graph for that given range and state what shape does the graph make and why? Pause and do that now.

Welcome back.

Let's see how we did.

Did we find that when x equals zero y equals four? When x equals one, y equals three.

Did we get that populated table of values, those coordinates and that line? That is the linear graph of y equals four minus x for the given range zero is less than or equal to x is less than or equal to four.

What shape does the graph make? It's a straight line.

Why? Because of the constant common difference.

For every change of positive one in x, we've got a change of negative one in y, hence a straight line graph or a linear graph.

The technology means we don't have to work out the coordinates for ourselves.

For example, if the problem was plot the linear graph of y equals 3x minus four for x values in the range of zero to positive four in step to 0.

5, that's a lot of y coordinates to work out.

We really don't want to have to work them all out manually, so we'll use the headers of the table to create a little efficiency for ourselves.

Instead of having the default Desmos headers, let's change these headers to x and 3x minus four.

Once you do that, the coordinates are automatically calculated for you and then they're automatically plotted.

All we have to do is click on that icon to turn on the line and there it is.

The linear function y equals 3x minus four for that given range.

Quick check you can do that.

I'd like you to plot the linear graph of y equals three minus 0.

5x for x values in the range of zero to positive three using steps of 0.

25.

Again, we really don't wanna have to calculate all those coordinates for ourselves, so you are gonna change those headers there to do this one a lot more efficiently.

Pause and give this a go now.

Welcome back.

Let's see how we did.

If you typed into the headers x and three minus 0.

5x, those coordinates should have been automatically calculated for you and automatically plotted for you.

Indeed, the only thing you had to do was click, hold and turn on the line and that would've given you the linear graph y equals three minus 0.

5x for the given range zero is less than or equal to x is less than or equal to three.

We can also plot negative x values.

For example, if we had to plot the linear graph y equals 3x plus one for x values in the range of negative three to positive three in steps of one, that's what our x coordinates would look like.

Negative three, negative two, negative one, et cetera.

Again, an efficiency here, if you type in negative three, negative two, negative one into the x column, if you press enter repeatedly, it'll auto autofill that column for you.

We can then type 3x plus one into the y column header and the coordinator calculated for us, plotted for us.

The only thing we had to do was turn on that line.

Quick check you can do that.

I'd like you to plot the linear graph of y equals three minus 2x for x values in the range of negative four to positive four.

Pause this video, see if you can do that now.

Welcome back.

Let's see how we did.

Did you get something like that? That table of values, those coordinates? Did you turn on the line? If so, that is the linear graph of y equals three minus 2x for the given range negative four is less than or equal to x is less than or equal to four.

The technology can plot these lines even more quickly for us.

We don't have to plot the individual coordinates from a table.

I could simply type in y equals nine minus 2x here.

If I do that, there it is.

The line y equals nine minus 2x but there's no restriction on the range.

This is the true nature of this graph.

It has no beginning and it has no end.

It goes on infinitely in both directions.

Quick check you can do that.

I'd like to plot linear graph y equals three minus 1/3 of x.

Use the forward slash key to write the fraction, but be sure to press the right arrow key to write the x after the fraction.

You don't want to write one over 3x, you want to write 1/3 of x.

Pause, give this a go now.

Welcome back.

See how we did.

Y equals three minus 1/3 of x should look just like that.

I can see a y intersect of three, a gradient of negative 1/3.

So we intersect the x-axis at the coordinate nine, zero.

Your graph should look just like mine.

This is the linear graph of y equals three minus 1/3x with no restriction on the range.

That said, we can limit the range when plotting linear graphs this way.

There is the same function, three minus 1/3 of x, but I'm gonna type curly brackets.

Negative one is less than or equal to x is less or equal to three, close curly brackets.

By typing that immediately after the function, it'll give the range these limits.

I type that immediately after where I've got my function and all of a sudden, I've limited the range.

This is the line graph of y equals three minus 1/3 of x for the given range negative one is less than or equal to x is less than or equal to three.

Quick check that you can repeat that skill on the same function y equals three minus 1/3x and you limit the range of your graph to zero is less than or equal to x is less than or equal to nine.

Pressing the equals key straight after typing the less than symbols will turn them from less than symbols into less than or equal to symbols.

And as a reminder, when I limited my range, that's what I typed immediately after my function.

So you can have a go at your graph and limit your range to x values zero is less than or equal to x is less than or equal to nine.

Pause and do that now.

Welcome back.

How did we do? Did we type that immediately after our function? Did our graph then look like that? If so, well done.

This is the linear graph y equals three minus 1/3 of x for the given range zero is less than or equal to x is less than or equal to nine.

Practise time now.

For question one, I'd like you to create a table of values to plot the linear graph of y equals 11 minus five x for x values in the range of negative two is less than or equal to x is less than or equal to three in steps of 0.

5.

Now, we know that creating a table of values like this is a slightly less efficient way of doing it, but it is a useful skill.

So I'd like you to create a table of values for this function in that range in those steps.

Pause and do that now.

For question two, without a table, plot the linear graph y equals seven minus x.

No restriction on range.

For part B without using a table, plot the linear graph of x plus y equals five.

Again, no restriction on range.

And then for part C, what do you notice about the shape of the two graphs? Write a sentence.

Pause, do those three things now.

Welcome back.

Feedback time.

Let's see how we did.

Table of values to plot that linear graph in the range of negative two up to positive three in steps of 0.

5.

Our table of values should look like that.

Our graph should have looked like that.

Pause, just check that yours matches mine.

For question two, without using a table, plot y equals seven minus x and x plus y equals five.

We get those two lines.

What do you notice about the shape of the two graphs? Write a sentence.

Lots of things we might have noticed.

You might have written both are linear graphs.

Well done.

You may also have said both have a gradient of negative one, making them parallel.

Why do they both have a gradient of negative one? Why are they parallel? Because we can rearrange x plus y equals five into the form y equals five minus x.

And look, x coefficients of negative one, gradients of negative one.

They're both parallel.

And you may have noticed is the x exponent of one in both equations that makes them linear.

Onto part two of the lesson now.

Plotting quadratic graphs.

Sounds exciting and it is.

Let's have a look.

There's many different types of equations and certain types of equations make graphs of certain shapes.

For example, let's explore a table of values for the equation y equals x squared for x values in the range of zero is less than or equal to x is less than or equal to five.

If we go in steps of one, we'll start with a table like that and we'll get y coordinates for when x equals zero, one equals zero squared.

When x equals one, y equals one squared and so on.

There's our table of values.

When we plot them, those coordinates look like this and we notice something.

These points are not in a straight line.

This is a non-linear graph.

So what do we do? Do we connect these points with straight lines? Should I do that? What do you think? Let's explore further.

What we can do is take shorter steps in our table to understand the shape of this function.

At the minute, I'm going up in steps of one in the x column.

What if I change that to steps of 0.

5? Did you see that difference? Steps of 0.

5 start to show us the graph's true shape.

Typing x and x squared into the column headers made that table populate a lot quicker and those coordinates automatically plot for me.

I'd like to check that you can just explore this a little further.

My x column from zero to positive five goes up in steps of 0.

5.

Could you create one using steps of 0.

25 to more clearly show the graph's true shape? Pause, make that table, create those coordinates.

Let's reveal the true shape of this graph.

Welcome back.

Let's see how we did.

Your table of values should have looked like that and your coordinates should look like this.

Steps of 0.

25 show that the graph of y equals x squared makes an upward curve of increasing gradient.

There's no straight lines to be seen here.

This is a curve.

Just like our linear graphs, we can input negative values of x too.

So what if we plotted y equals x squared and included some negative values? What if our range was negative four is less than or equal to x is less than or equal to positive four? Well, our table of value starts like this and then we populate it, remembering that when x equals negative four, y equals negative four squared.

That's negative four multiplied by negative four.

As I recall, that'll give us a positive, positive 16.

When x equals negative three, y equals negative three squared, positive nine.

If we keep going with that, we get that table of values.

What do you notice? Look at the table of values.

Pay particular attention to the y column.

Tell the person next to you what you notice or say it aloud to me at the screen.

Pause, do that now, see you in a moment.

Welcome back.

There's a lot you might have noticed.

Did you notice there is a repeated one in our y column, a repeated four in our y column, nine appears twice, 16 appears twice.

You could almost say we have a symmetry in those y values.

Why? Pardon the pun.

Because negative one squared is the same as positive one squared.

Negative two squared is the same as positive two squared and so on.

This does something beautiful when we go to plot these as coordinates.

That's what the coordinates look like.

The y-axis is like a line of symmetry reflecting each coordinate.

Quick check that you can do that now.

I'm gonna check that you can make a table of values for the equation y equals x squared.

You can use the same range, negative four to positive four, but I'd like you to move in steps of 0.

25.

So in your x column, type negative four, negative 3.

75, negative 3.

5, then start hitting enter, it will autopopulate for you.

A reminder, you don't wanna calculate all those y coordinates yourself, so change that header to read x squared.

Pause, see if you can make that table of values now.

Plot those coordinates as well.

See you shortly.

Welcome back.

Let's see how we did.

Did your table of values look like this? It's a very busy one, but a beautiful one.

We've got symmetry in that table of values.

Do your coordinates look like this? Well done.

Did you see we've got symmetry in those coordinates? The y-axis acting like a line of symmetry reflecting each coordinate.

You might wanna pause and just check your table of values looks just like mine, your coordinates look just like mine.

Welcome back.

The technology will quickly plot the infinite points of the equation y equals x squared for us very quickly.

We don't need that table of values.

We can just type y equals x squared in here.

Typing x to the power of two will generate the x squared in Desmos.

The power button is Shift + six on most keyboards.

Can you pause? See if you can find that.

See if you can type that in now.

Welcome back.

Let's see how we did.

Does your screen look like this? Y equals x squared.

The infinite points of y equals x squared and again, this line is a curve but it's going on infinitely in both directions.

The graph y equals x squared forms what we call a parabola.

This is a special curve, symmetrical and bowl-like in appearance.

Why is a parabola special? You may not know the answer to that question, but there's many answers.

Here's just one reason.

A parabola is unique in shape in that every point on the curve is equidistant from a single point called the focus and a straight line called the directrix.

You can see my point labelled focus and my straight line called directrix.

Every point on my parabola is equidistant from those two things.

A parabola is also uniquely useful.

Parabolic reflectors in car headlights mean that light emanating from a small bulb positioned at the focus point of the parabola reflects off in a focused parallel direction.

This is a unique property of parabolas.

What does that mean? Well, if you can imagine my focus point there is a light bulb, as the light shines out from that bulb and reflects off the parabolic reflector.

It's all reflecting in a focused parallel direction.

This enables a car headlight to have a very small bulb but generate an incredible lot of light in one focused direction.

This keeps motorists safe at night.

Well done, the parabola.

So the graph y equals x squared forms a parabola, a special symmetrical curve.

Will these graphs form other different special shapes? Y equals x squared plus two, y equals 2x squared, y equals x squared plus 2x.

Will they form a parabola or will it be a different special kind of shape? Pause this video, type them into Desmos and see if you can answer that question for yourself.

Welcome back.

Did we notice y equals x squared plus two look like that? Y equals 2x squared looks like so.

Y equals x squared plus 2x looks like that.

If we plot them all together, we'll notice they all form the same special shape, a parabola.

They might be slightly different in position and shape, but they all form a parabola.

Quick check you've got that.

The graph of this line y equal x squared plus 2x minus three, which you see here, forms a what? A straight line, a circle or a parabola? Pause, tell the person next to you.

Welcome back.

I do hope you said it wasn't a straight line, nor a circle.

Y equals x squared plus 2x minus three forms a parabola.

It's a special type of equation which forms a parabola.

What shape do each of these equations make? I'd like you to pause this video, type those into Desmos and compare the different shapes you see.

Welcome back.

Hope you enjoyed that moment of discovery.

Did you notice y equals 2x plus three makes a straight line, y equals 3x squared makes a parabola, y equals 2x cubed, you might not know what this is called.

It's called a cubic curve, but importantly, it's not a parabola.

What is unique about this equation that made it form a parabola? Can you spot it? The equation one equals 3x squared is a quadratic equation.

That's an equation whereby the highest exponent of the variable is two, 3x squared, 3x to the power of two.

It's that highest exponent of two that makes that graph a parabola.

Quick check you've got that.

Which of these equations are quadratic, therefore will form a parabola when plotted? A reminder, a quadratic equation is an equation whereby the highest exponent of the variable is two.

Pause.

Which ones will form a parabola? Which ones will not? See you in a moment.

Welcome back.

A will form a parabola.

The highest exponent of x is two.

Therefore, it's a quadratic equation that will form a parabola.

B will not.

The highest exponent of x is three, 5x to the power of three.

It's therefore not quadratic and will not form a parabola.

C will not form a parabola.

Highest exponent of x is three.

Not quadratic, will not form a parabola.

And D, well done, will form a parabola.

The highest exponent of x is two.

Therefore, it's a quadratic equation.

It will form a parabola.

Practise time now.

For question one, I'd like to categorise these equations into quadratic, linear and other.

Hint: Use Desmos to check if you're unsure.

You could type any of these into Desmos.

If the graph forms a parabola, it is quadratic.

Pause and categorise those equations now.

Question two.

I'd like you to use technology to plot the graphs of these quadratic equations.

They all form parabola.

I'd like you to spot the odd one out.

Pause, plot them.

Have a think about the odd one out.

See you soon.

Feedback time now.

For question one, I asked you to categorise some equations into quadratic, linear and other.

We should have noticed that A was linear, B was quadratic, C would be other.

It wasn't a parabola that you were seeing there.

It goes in the other category.

That's not x to the power of two, it's x the power of four.

Too high a power to be a quadratic.

D was a quadratic, therefore formed a parabola.

E was quadratic, F was quadratic.

G was linear, H was quadratic.

I was other.

J was linear, K was quadratic and L was other.

For question two, I asked you to plot these four quadratics and spot the odd one out.

You would've noticed that A and B form the same graph y because addition's commutative.

We're adding x squared to three or three to x squared.

They're the same thing.

Graph C, x squared minus three looked like that.

D, three minus x squared looked like that.

The odd one out.

Well, you could have said C.

You might have said C's the odd one out because it's the only one with a negative y-intercept, a perfectly fine argument.

I imagine many people looked at D and said D's the odd one out using language like it's upside down.

Whilst it's very different, it is still a parabola.

It looks upside down, but we tend to say it opens downwards or is N shaped.

Importantly, we still have a quadratic equation and a parabola.

Here we are at the end of the lesson now.

We can appreciate that different types of equations give rise to different graph shapes.

Linear equations like y equals five plus 2x form a straight line, whereas quadratic equations like y equals x squared plus three form a parabola.

I hope you've enjoyed exploring graphs as much as I have today and I look forward to seeing you again soon for more mathematics.

Goodbye for now.