Teaching parametric equations using graphing technology

Teaching parametric equations
using graphing technology
The session will start by looking at problems which help
students to see that parametric equations are not there to
make life difficult but are important and give rise to some
beautiful curves. Then we will look at ways in which
graphing technology (calculators, GeoGebra and
Autograph) can help students make sense of parametric
curves for themselves.
Suitable for all teachers who have taught parametric
equations in A level Mathematics.
Teaching parametric equations
using graphing technology
• Witch of Agnesi and cycloids: learn the calculator
functionality with some nice maths.
• Cards activity: thinking how calculators can deepen
understanding, supporting learning in class.
• Past exam question and 2017 SAMs questions: thinking
how calculators can support exam technique.
• Introducing parametrics using GeoGebra.
• A few investigations.
The Witch of Agnesi
Maria Agnesi,
1718 - 1799
https://www.geogebra.org/m/F9xJzEqU
The Witch of Agnesi
There has been much argument over the reason why the curve is called a
'witch'.
In 1718 Guido Grandi (1671-1742, an Italian Jesuit who worked on
geometry and hydraulics) gave it the Latin name 'versoria' which means
'rope that turns a sail' and he so named it because of its shape. Grandi
gave the Italian 'versiera' for the Latin 'versoria' and indeed Agnesi quite
correctly states in her book that the curve was called 'la versiera'.
John Colson translated Agnesi's Instituzioni analitiche ad uso della gioventù
italiana into English [and] mistook 'la versiera' for 'l'aversiera' which means
'the witch' or 'the she-devil'.
http://www-history.mcs.st-and.ac.uk/
Cycloids
x  aT  a sin T
y  a  a cos T
https://www.geogebra.org/m/EMaY9cxR
MEI Core 4. June 2013 Qn 7(part)
MEI Core 4. June 2013 Qn 7(part)
MEI Core 4. June 2013 Qn 7(part)
MEI Core 4. June 2012. Qn 7(part)
MEI Core 4. June 2012. Qn 7(part)
MEI Core 4. June 2012. Qn 7(part)
https://www.geogebra.org/m/UfDqG6Ad
MEI Conference 2016
Teaching parametric
equations using
graphing technology
Bernard Murphy
[email protected]
The Witch of Agnesi and the Casio fx-CG20
The circle has radius 1 and centre  0,1 . The point A moves along the line y  2 . The line
OA, which makes an angle T with the y -axis, meets the circle at B. Lines BP and PA are
parallel to the axes. What is the locus of P?
-------------------------------
First reset your calculator to factory settings:pazywqd
1. Add a new Graphs screen: p5
2. Set the graph type to Parametric: ee
3. Draw the graph x  2 tan T , y  2 cos 2 T :
2kfl2(jf)slu
4. Experiment with the V-Window Le to set appropriate ranges for X, Y and
T. After each one use l and end withd.
5. By eliminating the parameter, find the Cartesian equation of this curve. See if
you’re right by entering this on your calculator, first setting the graph type to
Cartesian: deq
Cycloids
A wheel rolls along a flat surface without slipping. A point P is on the circumference of the
wheel. What is the equation of the path of P?
Let the point P on the rim start at the origin, O. The diagram shows the position of the wheel
after it has rolled a distance to the right.
Since there is no slipping, the horizontal distance moved along the road, OA, must be the
same as the distance PA on the wheel rim.
OA = PA = aT (where T is measured in radians).
When C has coordinates  aT , a  , P has coordinates  aT  a sin T , a  a cos T  .
So the path of P can be described by parametric equations:
x  aT  a sin T ,
y  a  a cos T
Investigate the locus of other points along the radius.
dx
1
 2 2
dt
t
y  3t 2
dy
 6t
dt
t  1, gives the
point (3 ,3)
At t  1,
dy
6
dx
x4 t
dx 2

dt
t
y  ln t
dy 1

dt t
t  1, gives the
point (4 ,0)
At t  1,
dy 1

dx 2
x  t2
dx
 2t
dt
y  t3
dy
 3t 2
dt
t  1, gives the
point (1 ,1)
At t  1,
dy 3

dx 2
x t
dx
1
dt
y  t2 1
dy
 2t
dt
t  1, gives the
point (1 ,2)
At t  1,
dy
2
dx
x  4t  t 2
dx
 4  2t
dt
y  2t
dy
 1
dt
t  1, gives the
point (5 ,1)
At t  1,
dy
1

dx
6
x  3t  1
dx
3
dt
dy
2
 3
dt
t
t  1, gives the
point (4 ,1)
At t  1,
dy
2

dx
3
1
x   4t
t
dx
1
 4 2
dt
t
dy
1
 2
dt
t
t  1, gives the
point (5 ,2)
At t  1,
dy
1

dx
3
x  2t 
1
t
y
1
t2
1
y  1
t
Tangents to parametric curves
(Adapted from a resource from http://www.mei.org.uk/calculators )
1. Add a new Graphs screen: p5
2. Set Derivative: On and Angle: Radians:
LpNNNNNNqNNNNwd
3. Set the graph Type to Parametric:
functions here.)
ee (You might want to delete existing
4. Use V-Window to set the range of T to Tmin: -3, Tmax: 3, pitch: 0.05
LeNNNNNNNn3l3l0.05ld
5. Draw the graph x  t 2 , y  t 3 . X1t=T², Y1t=T³: fslf^3lu
6. Add the tangent at the point (Sketch > Tangent): rw
Use !/$ to move the position of the point on the curve or choose a value using f.
Questions
dy dx
dy
,
and
?
dx dt
dt

What is the relationship between

Can you verify this algebraically for some other parametric curves?
Problem (Check your answer by plotting the graph and the tangent on your calculator)
Find the coordinates of the points on the curve x  2t  cos t , y  sin t ,  2  0  2 for
which the tangent to the curve is parallel to the x axis.
Further Tasks

Explore how you can find the equation of the tangent to a parametric curve at a point.

Describe how to find the tangent to a parametric curve that passes through a specific
point that is not on the curve.
-----------------------------------First reset your calculator to factory settings:pazywqd
1. Add a new Graphs screen: p5
2. Set the graph type to Parametric: ee
3
3. Draw the graph x  2t  1, y  4t  7  :
t
2f-1l4f-7+3Mflu
4. Experiment with the V-Window Le to set appropriate ranges for X, Y
and T. After each one hit l and end with d.
5. Set the graph type to Cartesian deqand either eliminate the
parameter to find the Cartesian equation of this curve or experiment using the
modify command:
(2fs+aff+ag)M(f+1)ly
6. Using the cursors change the values of A and B to view the resulting graph.
1. Add a new Graphs screen: p5
2. Set Derivative: On and Angle: Radians:
LpNNNNNNqNNNNwd
3. Set the graph Type to Parametric:
functions here.)
ee (You might want to delete existing
4. Use V-Window to set appropriate ranges for x, y and T using Le the
N
cursor, l after each entry and d to finish.
5. Enter x  2 cos t , y  sin t and u to draw.
6. Add the tangent at the point (Sketch > Tangent): Lrw and use !/$ to
move the position of the point on the curve or choose a value using f.
June 2012. MEI Core 4. Question 7(part)
June 2013. MEI Core 4. Question 7(part)
Investigation 1
Imagine two points, P and Q , on a unit circle.
They start from 1,0  and go at constant speeds in the
same direction round the circle but Q goes twice as fast
as P .
Now think about the chord PQ . Its length is changing
continuously. What path is followed by the midpoint, M ,
of PQ ?


What if the ratio of the speeds of P and Q was different?
What if P and Q went in different directions?

What path is followed by the point N on the chord where PN 
1
PQ ?
3
Investigation 2
Point P is on the line y  1 .
A circle centred on P touches the unit circle as
shown.
The radius PQ is parallel to the x axis.
As P moves along the line y  1 , what is the path
traced out by point Q ?
Investigation 3
Using your graphing calculator, investigate curves with parametric equations
x  3cos t  cos kt ,
y  3sin t  sin kt
For which values of k do you get a dimple?...a cusp?...crossover points?
Find the coordinates of points where the tangent is parallel to an axis.
What other properties could you explore?