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.)

Begin by showing a point and its dilated image, and describing how the dilation behaves.

- 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*D*(_{C,s}*x*) as “the dilation of*x*about*C*by scale factor*s*.” **Drag***x*up. How does*D*(_{C,s}*x*) move? Drag*x*left. How does*D*(_{C,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 point.
**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*D*(_{C,s}*x*) come together is called a*fixed point*of the function.

- 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 it reappear. **Drag***x*straight up. Which way does*D*(_{C,s}*x*) go, and how fast? Drag*x*left. Which way does*D*(_{C,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.

- Click the pointer to go to page 2 of the sketch.
- Drag
*x*and observe the behavior of*D*(_{C,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 restricted domain.**- Press
*Animate x*to move*x*around its domain. As*x*varies,*D*(_{C,s}*x*) traces out the range. **How does the range compare to the restricted domain? What features of the domain and range are similar, and what features are different?**

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 function expert!

- 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.