Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To...
-
Upload
marylou-jacobs -
Category
Documents
-
view
215 -
download
1
Transcript of Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To...
![Page 1: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/1.jpg)
x5sin3 2
Extending what you know…
![Page 2: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/2.jpg)
dxx cos3
![Page 3: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/3.jpg)
dx 3cos5cos xx
![Page 4: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/4.jpg)
dx 3cos5sin xx
![Page 5: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/5.jpg)
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
![Page 6: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/6.jpg)
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.
![Page 7: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/7.jpg)
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.
![Page 8: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/8.jpg)
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.
![Page 9: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/9.jpg)
Examples
Find the equation in Cartesian form from the parametric equations x = t + 1 and y = t2
![Page 10: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/10.jpg)
Examples
Define x=3cost and y=5sint in Cartesian form.
![Page 11: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/11.jpg)
Card Match
Pair up cards
into
Parametric and matchingCartesian form
![Page 12: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/12.jpg)
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
![Page 13: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/13.jpg)
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
![Page 14: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/14.jpg)
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
![Page 15: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/15.jpg)
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
![Page 16: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/16.jpg)
Parametric Equations
sin3,cos2 yx 22 x and
33 y
Symmetry now completes the diagram.
x
x
x90
x
0
![Page 17: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/17.jpg)
Parametric Equations
sin3,cos2 yx 22 x and
33 y
Symmetry now completes the diagram.
x
x
x90
x
0
![Page 18: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/18.jpg)
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).
![Page 19: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/19.jpg)
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
![Page 20: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/20.jpg)
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.
![Page 21: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/21.jpg)
Card Match
Pair up cards
into
graphs and matchingParametric equations
![Page 22: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/22.jpg)
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 ).
![Page 23: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/23.jpg)
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.
![Page 24: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/24.jpg)
Parametric Equations
tan2,sec4 yxSketch:
1416
22
yx
or
A hyperbola
Asymptotes
![Page 25: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/25.jpg)
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
![Page 26: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/26.jpg)
Parametric Equations
tytx
3,3 9xySketch
:or
A rectangular hyperbola.
Asymptotes
![Page 27: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/27.jpg)
Plenary
Pretty Parametric Equations
1) x=Sin 45t y=sin22.5t 2) x=sin22.5t y=sin11.25t
![Page 28: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/28.jpg)
Parametric Equations
notes
![Page 29: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/29.jpg)
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
![Page 30: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/30.jpg)
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
![Page 31: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/31.jpg)
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.
![Page 32: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/32.jpg)
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
![Page 33: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/33.jpg)
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.
![Page 34: Extending what you know…. Parametric Equations Aims: To know how what a parametric equation is. To be able to draw a parametric curve given the.](https://reader030.fdocuments.us/reader030/viewer/2022032414/56649ef35503460f94c04f7c/html5/thumbnails/34.jpg)
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).