Parametric Equations

Module C4

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Parametric Equations

θθ sin3,cos3 == yx

The Cartesian equation of a curve in a plane is an

equation linking x and y.

Some of these equations can be written in a way

that is easier to differentiate by using 2 equations,

one giving x and one giving y, both in terms of a 3rd

variable, the parameter.

Letters commonly used for parameters are s, t and

θ. ( θ is often used if the parameter is an angle. )

e.gs. tytx 4,2 2

==

Parametric Equations

Converting between Cartesian and Parametric forms

We use parametric equations because they are

simpler, so we only convert to Cartesian if asked to

do so !

e.g. 1 Change the following to a Cartesian equation

and sketch its graph:

tytx 4,2 2

==

Solution: We need to eliminate the parameter t.

We substitute for t from the easier equation:

⇒= ty 4

4

y

t =

Subst. in :2 2

tx =

2

4

2

=⇒

y

x

8

2

y

x =⇒

Parametric Equations

The Cartesian equation is

8

2

y

x =

We usually write this as xy 82

=

Either, we can sketch using a graphical calculator

with

xy 8±=

and entering the graph in 2 parts.

Or, we can notice that the equation is quadratic with

x and y swapped over from the more usual form.

Parametric Equations

The sketch is

The curve is called a parabola.

xy 82

=

tytx 4,2 2

==Also, the parametric equations

show that as t increases, x increases faster than y.

Parametric Equations

e.g. 2 Change the following to a Cartesian equation:

θθ sin3,cos3 == yx

Solution: We need to eliminate the parameter θ.

BUT θ appears in 2 forms: as and

so, we need a link between these 2 forms.

θcos θsin

Which trig identity links and ?θcos θsin

ANS: 1sincos 22

≡+ θθ

To eliminate θ we substitute into this expression.

Parametric Equations

Since we recognise the circle in Cartesian form,

it’s easy to sketch.

However, if we couldn’t eliminate the parameter or

didn’t recognise the curve having done it, we can

sketch from the parametric form.

If you are taking Edexcel you may want to skip this

as you won’t be asked to do it.

SKIP

Parametric Equations

Solution:

Let’s see how to do it without eliminating the

parameter.

We can easily spot the min and max values of x and y:

22 ≤≤− x and 33 ≤≤− y

( It doesn’t matter that we don’t know which angle

θ is measuring. )

For both and the min is −1 and the

max is +1, so

θcos θsin

θθ sin3,cos2 == yx

e.g. Sketch the curve with equations

It’s also easy to get the other coordinate at each

of these 4 key values e.g. 002 =⇒=⇒= yx

θ

Parametric Equations

⇒== θθ sin3,cos2 yx 22 ≤≤− x and 33 ≤≤− y

We could draw up a

table of values finding

x and y for values of

θ but this is usually

very inefficient. Try

to just pick out

significant features.

x

x

x

90=θ

x

0=θ

Parametric Equations

⇒== θθ sin3,cos2 yx 22 ≤≤− x and

x

33 ≤≤− y

This tells us what happens to x and y.

90

Think what happens to and as θ increases

from 0 to .

θcos θsin

We could draw up a

table of values finding

x and y for values of

θ but this is usually

very inefficient. Try

to just pick out

significant features.

x

x

x

90=θ

x

0=θ

Parametric Equations

⇒== θθ sin3,cos2 yx 22 ≤≤− x and

x

Symmetry now

completes the diagram.

33 ≤≤− y

This tells us what happens to x and y.

90

Think what happens to and as θ increases

from 0 to .

θcos θsin

x

x

x

90=θ

x

0=θ

Parametric Equations

θθ sin3,cos2 == yx

So, we have the parametric equations of an ellipse

( which we met in Cartesian form in Transformations ).

The origin is at the

centre of the ellipse.

x

x

x

x

Ox

Parametric Equations

You can use a graphical calculator to sketch

curves given in parametric form. However, you

will have to use the setup menu before you enter

the equations.

You will also have to be careful about the range

of values of the parameter and of x and y. If

you don’t get the right scales you may not see

the whole graph or the graph can be distorted

and, for example, a circle can look like an

ellipse.

By the time you’ve fiddled around it may have

been better to sketch without the calculator!

Parametric Equations

The following equations give curves you need to

recognise:

θθ sin,cos ryrx ==

atyatx 2,2

==

)(sin,cos babyax ≠== θθ

a circle, radius r, centre

the origin.

a parabola, passing through

the origin, with the x-axis as

an axis of symmetry.

an ellipse with centre at the

origin, passing through the

points (a, 0), (−a, 0), (0, b), (0, −b).

Parametric Equations

To write the ellipse in Cartesian form we use the

same trig identity as we used for the circle.

)(sin,cos babyax ≠== θθSo, for

use 1sincos 22

≡+ θθ

12

2

2

2

=+

b

y

a

x

⇒

1

22

=

+

b

y

a

x

⇒

The equation is usually left in this form.

Parametric Equations

There are other parametric equations you might be

asked to convert to Cartesian equations. For example,

those like the ones in the following exercise.

Exercise

θθ tan2,sec4 == yx

t

ytx

3

,3 ==

( Use a trig identity )

1.

2.

Sketch both curves using either parametric or Cartesian

equations. ( Use a graphical calculator if you like ).

Parametric Equations

Solution:

θθ tan2,sec4 == yx1.

Use θθ 22

sectan1 ≡+

22

42

1

=

+⇒

xy

164

1

22

xy

=+⇒

We usually write this in a form similar to the

ellipse:

1

416

22

=−

yx

Notice the minus sign. The curve is a hyperbola.

Parametric Equations

The following slides contain repeats of

information on earlier slides, shown without

colour, so that they can be printed and

photocopied.

For most purposes the slides can be printed

as “Handouts” with up to 6 slides per sheet.

Parametric Equations

θθ sin3,cos3 == yx

The Cartesian equation of a curve in a plane is an

equation linking x and y.

Some of these equations can be written in a way

that is easier to differentiate by using 2 equations,

one giving x and one giving y, both in terms of a 3rd

variable, the parameter.

Letters commonly used for parameters are s, t and

θ. ( θ is often used if the parameter is an angle. )

e.gs. tytx 4,2 2

==

Parametric Equations

Converting between Cartesian and Parametric forms

We use parametric equations because they are

simpler, so we only convert to Cartesian if asked to

do so !

e.g. 1 Change the following to a Cartesian equation

and sketch its graph:

tytx 4,2 2

==

Solution: We need to eliminate the parameter t.

Substitution is the easiest way.

⇒= ty 4

4

y

t =

Subst. in :2 2

tx =

2

4

2

=⇒

y

x

8

2

y

x =⇒

Parametric Equations

The Cartesian equation is

8

2

y

x =

We usually write this as xy 82

=

Either, we can sketch using a graphical calculator

with

xy 8±=

and entering the graph in 2 parts.

Or, we can notice that the equation is quadratic with

x and y swapped over from the more usual form.

Parametric Equations

e.g. 2 Change the following to a Cartesian equation:

θθ sin3,cos3 == yx

Solution: We need to eliminate the parameter θ.

BUT θ appears in 2 forms: as and

so, we need a link between these 2 forms.

θcos θsin

To eliminate θ we substitute into the expression.

1sincos 22

≡+ θθ

Parametric Equations

The following equations give curves you need to

recognise:

θθ sin,cos ryrx ==

atyatx 2,2

==

)(sin,cos babyax ≠== θθ

a circle, radius r, centre

the origin.

a parabola, passing through

the origin, with the x-axis an

axis of symmetry.

an ellipse with centre at the

origin, passing through the

points (a, 0), (−a, 0), (0, b), (0, −b).