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The name of the function, "Witch of Agnesi", is a complete
misnomer. Many people who have heard of the function are not aware that
Agnesi was a woman (Maria Gaetana Agnesi, 1718-1799) and ironically, was
not the discoverer of the function. The French mathematician Pierre de
Fermat is said to have written about the function in 1665, almost a hundred
years before Ms. Agnesi ever did. Furthermore, the word witch in
its title is the result of an inaccurate English translation. It has been
recorded in 1703 that another mathematician, Guido Grandi, named the function
Versaria , meaning "turning in every direction". In the course
of time the word versaria took on another meaning. The Latin words
adversaria, by aphaeresis, and versaria, signify
a female that is contrary to God. Gradually the function versaria
came to be known in English as "the witch".
Part 1:
"The Witch" is a function, so we would like to graph it on a
coordinate system. Define a Coordinate System in your sketch window.
To define a coordinate
system:
- Select Define Coordinate System, under the Graph menu.
- You may drag the point (1,0) along the x-axis to define the size
of the scaling unit.
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- Construct a circle centered at the point (0,2), with a radius
of 2 units. Remember, to construct a circle you can use the Circle
by Center+Point command under the Construct menu.
To plot
a point on the coordinate system:
- Select Plot Points, under the Graph menu.
- Enter the x and y coordinates of a point you would like to
be plotted, and click the Plot button.
- Click the Done button.
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- Construct a line tangent to the circle, above and parallel to
the x-axis.
- Construct an arbitrary point on the circle, and drag this point around
the circle until it lies in the first quadrant. Change its color to
green.
To change
the color of an object in your sketch window:
- Select the object.
- Select Color under the Display menu.
- A palette of colors will appear. Click the mouse on a color
of your liking.
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- Construct a Ray from the origin through the green point.
- Find the point of intersection of the tangent line and the ray. Change
its color to blue.
- Using the Line Style command under the Display menu,
construct a dashed perpendicular line through the blue point to the
x-axis; and a dashed perpendicular line through the green point to the
y-axis.
- Construct the point of intersection between the two dashed lines,
and color it red.
The function formed by tracing the red point as the green point travels
around the circle is called the Witch of Agnesi.
- Drag the green point around the circle, paying particular attention
to the movement of the red point. Make a conjecture about the general
shape of the function. On paper, draw a quick sketch illustrating
your ideas.
Part 2:
- We would like to trace the red point as the green point travels
around the circle. Create a button to animate the green point traveling
one way around the circle slowly. (Instructor
Note: A
Sketchpad file illustrating the construction of the Witch of Agnesi
has been saved as witch.gsp.)
To identify
which point to trace:
- Select the point.
- Choose Trace Point under the Display menu.
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To create
an animation button:
- Select the object you wish to animate.
- Choose Action Button under the Edit menu, and
then Animation.
- In the Properties of Action Button window, make travel selections for
your object (i.e., its direction and speed).
- Click on the Label tab to rename your action button,
and click OK.
- An Animate button will appear in your sketch window.
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- Your action button toggles on and off by clicking on it. Activate your
animation button, and observe the traced intersection point.
- How does the resulting function compare with your previous conjecture?
Reconcile any differences.
- While the red point is being traced, there is a difference in the
density of the points being plotted. Explain why this occurs.
Part 3:
- Describe the characteristics of an asymptote. Does this function have
any asymptotes? Play around with the scale of your graph. How does changing
the graph’s scale influence your perception of the location of the function’s
asymptotes?
- Describe the characteristics of an inflection point. Does the function
have any inflection points? Explain and give the approximate coordinates
of each.
- Derive the equation of the function such that the red point will be
all (x,y) pairs which satisfy the equation. Hint: Look for a
pair of similar triangles and equate ratios of their sides.
| The derivation
of the equation of The Witch of Agnesi:
Go back to the construction on the first page of this activity
and equate the ratios of the sides of the similar triangles.
Since the radius of the circle is 2, F = (xc, y)
and H = (x,y), then
EA = 4
IA = y (That is, the y-coordinate of both F and H)
EG = x (That is, the x-coordinate of H)
IF = xc (That is, the x-coordinate of F)
Substituting these values in the above ratio equation, we have
From the equation of our circle xc2 +
(y-2)2 = 22 we know that xc =
Ö 22- (y-2)2.
We use (y-2) instead of y here to account for the displacement
of the circle in our figure, whose center is not at the origin.
Substituting this value of xc in the previous equation
we get
Squaring both sides of the preceeding equation and doing some
other simplification, we get
and finally x2y = 16(4-y) or  .
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- Once the equation has been derived, mathematically verify the existence
of both your conjectured asymptote(s) and inflection point(s).
Extensions:
- Derive the equation for the Witch of Agnesi using a polar coordinate system.
- Think of other investigations/activities that could be explored with this function. Write a summary of your ideas, and be prepared to present them to the class.
- Research the life of Maria Gaetana Agnesi (or another female mathematician) and prepare a presentation about her life and mathematics.
- Create another construction that uses the animation button in a useful
manner. Write up a description about the construction and demonstrate
your construction to the class.
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