The Jactus

Question 1

A Jactus is a mythical plant similar to a cactus, except a Jactus grows at a very peculiar rate. In fact, a trait common to all Jactus plants is that they will double in height every week. Use the slider in the Geogebra applet to experiment. Verify that the function  gives the height in inches of a Jactus plant after  weeks if the initial height was  inches. For example, for a 3-inch Jactus, the function  generates the following table of values: 
week height 
(you try)
(you try)

Question 2

Suppose we have a 3-inch Jactus plant (on Day 0).  How tall is the Jactus on day 7? Be careful: Day 7 does not mean that . Instead, Day 7 corresponds to . Why? Notice that the Jactus grows at a continuous rate rather than jumping up in one large growth-spurts at the end of the first week. Under that assumption:    a) How tall is a 3-inch Jactus at the start of Day 1? (Hint: What fraction of a week has elapsed?)    b) How tall is the 3-inch Jactus on day 10? (Hint: What is w now?) 

Question 3

We will need to be able to describe how the height of a Jactus changes over time. Example: If the Jactus's height increases from 4 inches to 5 inches over a certain timespan, we can describe the change in two ways:
  • We might say the height increased by 1 inch (this is an additive relationship).
  • Or we might say the height increased by a factor of 1.25 (a multiplicative relationship). We are reporting the scale factor. We can calculate it simply enough: scale factor = (new height) divided by (previous height).
Consider again the 3-inch Jactus plant from Question 2. (a) From an additive perspective, did the Jactus have a larger change in height on Day 0 or on Day 14? (b) And now from a multiplicative perspective:
  • What is the scale factor for the change in height on Day 0?
  • What is the scale factor for the change in height on Day 14? 
  • What is the scale factor for the change in height on Day 10?
What do you notice? 

Question 4

Now consider a 4-inch Jactus plant (at least on Day 0).   a) How many inches does the 4-inch Jactus grow during the first day?    b) Compare the 3- and 4-inch Jactus plants. Which has a larger change in height on its first day?    c) Compute the scale factor for the change in height on Day 0 for the 4-inch Jactus.   d) Compare with what you found for the 3-inch Jactus. What do you notice?