x5sin3 2

Extending what you know…

dxx cos3

dx 3cos5cos xx

dx 3cos5sin xx

Parametric Equations

Aims:•To know how what a parametric equation is.

•To be able to draw a parametric curve given the equation

•To be able to convert a parametric equation to a Cartesian equation

Parametric Equations

• You are used to equations in Cartesian form.• Eg y=f(x)• Parametric equations are where x and y are defined by some other variable.• Usually the letter t, y=f(t) x=g(t)• Sometimes this is more useful and allows you to define more complex graphs.

Using Your Calculator• You can plot parametric equations on

your calculator.• Enter graphing mode • Press (F3) for TYPE• And press (F3) again for Parm• This will then change the display to

show• Xt1= Yt1=• And you can enter the equations for

x and y in terms of t.

Eliminating the Parameter

• Sometimes it is useful to eliminate the parameter.

• To do this you must rearrange the equations to be able to eliminate t.

• When the parametric equations contain sin and cos you may need to use trig identities to get an equation in a more useful form.

Examples

Find the equation in Cartesian form from the parametric equations x = t + 1 and y = t2

Examples

Define x=3cost and y=5sint in Cartesian form.

Card Match

Pair up cards

into

Parametric and matchingCartesian form

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 q is measuring. )For both and the min is -1 and the max is +1, so

cos sin

sin3,cos2 yx 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

Parametric Equations

sin3,cos2 yx 22 x and

33 y

We could draw up a table of values finding x and y for values of q but this is usually very inefficient. Try to just pick out significant features.

x

x

x90

x

0

Parametric Equations

sin3,cos2 yx 22 x and

x

33 y

This tells us what happens to x and y.

90Think what happens to and as q increases from 0 to .

cos sin

We could draw up a table of values finding x and y for values of q but this is usually very inefficient. Try to just pick out significant features.

x

x

x90

x

0

Parametric Equations

sin3,cos2 yx 22 x and

x

Symmetry now completes the diagram.

33 y

This tells us what happens to x and y.

90Think what happens to and as q increases from 0 to .

cos sin

x

x

x90

x

0

Parametric Equations

sin3,cos2 yx 22 x and

33 y

Symmetry now completes the diagram.

x

x

x90

x

0

Parametric Equations

sin3,cos2 yx 22 x and

33 y

Symmetry now completes the diagram.

x

x

x90

x

0

Parametric Equations

The following equations give curves you need to recognise:

sin,cos ryrx

atyatx 2,2

)(sin,cos babyax

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

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

xO

x

Parametric Equations

To write the ellipse in Cartesian form we use the same trig identity as we used for the circle.

)(sin,cos babyax So, for

use 1sincos 22

12

2

2

2

b

y

a

x

122

b

y

a

x

The equation is usually left in this form.

Card Match

Pair up cards

into

graphs and matchingParametric equations

Parametric EquationsThere are other parametric equations you might be asked to convert to Cartesian equations. For example, those like the ones in the following exercise.Exercise

tan2,sec4 yx

tytx

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:

tan2,sec4 yx1.

Use 22 sectan1 22

421

xy

1641

22 xy

We usually write this in a form similar to the ellipse:

1416

22

yx

Notice the minus sign. The curve is a hyperbola.

Parametric Equations

tan2,sec4 yxSketch:

1416

22

yx

or

A hyperbola

Asymptotes

Parametric Equations

tytx

3,3

( Eliminate t by substitution. )

2.

Solution: 3

3x

ttx

t

y3

Subs. in

xy

9

9 xy

3

3x

y

The curve is a rectangular hyperbola.

xx 33 33

Parametric Equations

tytx

3,3 9xySketch

:or

A rectangular hyperbola.

Asymptotes

Plenary

Pretty Parametric Equations

1) x=Sin 45t y=sin22.5t 2) x=sin22.5t y=sin11.25t

Parametric Equations

notes

Parametric Equations

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

e.gs. tytx 4,2 2

Parametric Equations

Converting between Cartesian and Parametric formsWe 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 44

yt

Subst. in

:2 2tx 2

42

y

x8

2yx

Parametric Equations

The Cartesian equation is 8

2yx

We usually write this as

xy 82 Either, we can sketch using a graphical calculator with xy 8and 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: sin3,cos3 yx

Solution: We need to eliminate the parameter q.

BUT q appears in 2 forms: as andso, we need a link between these 2 forms.

cos sin

To eliminate q we substitute into the expression.

1sincos 22

Parametric Equations

cos3

x

9sin

22 y

9

cos2

2 x

sin3

y

1sincos 22 199

22

yx

922 yxMultiply by 9:

becomes

sin3,cos3 yxSo,

N.B. = not

We have a circle, centre (0, 0), radius 3.

Parametric Equations

The following equations give curves you need to recognise:

sin,cos ryrx

atyatx 2,2

)(sin,cos babyax

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).