Dilation Challenges: The Dilation Family
In this activity you will compare the motion of a point to the motion
of its dilated image. (List items in bold below
contain questions for you to answer, as you work to analyze and
understand your results.)
Dilate a Point
Begin by showing a point and its dilated image, and describing how the
- Show independent variable x and drag it around. This is
the independent variable.
- Show the center point C and the scale parameter s.
- Dilate x and drag x again. The blue point
depends on x, so we call it the dependent variable.
- Show the dependent variable’s label.
You can read DC,s(x)
as “the dilation of x about C by scale factor s.”
- Drag x up. How does DC,s(x)
move? Drag x left. How does DC,s(x)
move? In what direction, and how fast?
- Turn on tracing for both variables. Drag independent point x
to trace out an interesting shape.
- Describe the traced shapes. How are they similar, and how
are they different? Consider position, size, angle, and anything
else you think of. In your answer include a drawing or screen
capture of your two traced shapes.
- Erase the traces and trace a new shape that goes through the center
- What happened when you went through the center of dilation?
Describe these traced shapes, and include a drawing or screen
capture in your answer. Were there any fixed points? If so, where?
A location where x
and DC,s(x) come together is called a
fixed point of the function.
Use a Different Scale Factor
- Click parameter s and change its value to 2.00.
If the dependent variable
disappears, you may have to drag x or C to make
- Drag x straight up. Which way does DC,s(x)
go, and how fast? Drag x left. Which way does DC,s(x)
go, and how fast?
- Drag point x to make a shape.
- How are these shapes different from the shapes you made when
you dilated by 0.50? In your answer describe the traced shapes and
include a drawing or screen capture.
- Can you drag the two points together to find a fixed point?
If so, where is it?
- Change s to 1.00, erase the traces, and then drag
x. What happens? Where are the fixed points?
- Change s to –1.00, erase the traces, and then drag
x. What happens when the scale factor is negative?
- Predict what will happen (a) if s is –2.00, (b) if
s is very small, and (c) if s = 0. Explain and
draw your predictions first, before you test them.
Restrict the Domain
Now you’ll restrict point x
to a polygon and observe the
effect on DC,s
- Click the pointer to go to page 2 of the sketch.
- Drag x and observe the behavior of DC,s(x).
- Restrict x to the polygon by pressing Restrict x to
polygon. Drag x again.
- Describe the function’s domain. In other words, where can
you drag the independent variable? A domain like this is called a
- Press Animate x to move x around its domain. As
x varies, DC,s(x) traces out
- How does the range compare to the restricted domain? What
features of the domain and range are similar, and what features are
Now you’ll have a chance to try some dilation challenges. Some will be
harder than others; if you accomplish them all, you'll be a dilation
- Click the pointer to go to the first challenge.
- For each challenge page, describe clearly how you solved it, and
include a drawing or screen capture. Describe any shortcuts you
invented to make the challenge easier.