© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 47: Solving Differential Equations.
41: Parametric Equations “Teach A Level Maths” Vol. 2: A2 Core Modules.
-
Upload
jamari-overley -
Category
Documents
-
view
225 -
download
3
Transcript of 41: Parametric Equations “Teach A Level Maths” Vol. 2: A2 Core Modules.
41: Parametric 41: Parametric Equations Equations
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
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 . ( 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.
We substitute for t from the easier equation:
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
The sketch is
The curve is called a parabola.
xy 82
tytx 4,2 2 Also, the parametric equationsshow that as t increases, x increases faster than y.
Parametric Equations
e.g. 2 Change the following to a Cartesian equation: sin3,cos3 yx
Solution: We need to eliminate the parameter .
BUT appears in 2 forms: as andso, we need a link between these 2 forms.
cos sin
Which trig identity links and ?
cos sin
ANS: 1sincos 22 To eliminate we substitute into this
expression.
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
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.
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 is measuring. )For both and the min is 1 and the max is 1, so
cos sin
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
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 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 increases from 0 to .
cos sin
We could draw up a table of values finding x and y for values of 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 increases from 0 to .
cos sin
x
x
x90
x
0
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
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:
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
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.
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
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
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
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 . ( 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 .
BUT appears in 2 forms: as andso, we need a link between these 2 forms.
cos sin
To eliminate 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).