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.[br][br]Use the slider in the Geogebra applet to experiment. Verify that the function [math]h\left(w\right)=h_0\cdot2^w[/math] gives the height in inches of a Jactus plant after [math]w[/math] weeks if the initial height was [math]h_0[/math] inches.[br][br]For example, for a 3-inch Jactus, the function [math]h\left(w\right)=3\cdot2^w[/math] generates the following table of values: [br][table][tr][td][u]week [/u][math]w[/math][u][/u][/td][td][u]height [/u][math]h\left(w\right)[/math][/td][/tr][tr][td][math]0[/math][/td][td][math]3\cdot2^0=3[/math] in[/td][/tr][tr][td][math]1[/math][/td][td][math]3\cdot2^1=6[/math] in[/td][/tr][tr][td][math]2[/math][/td][td][math]3\cdot2^2=12[/math] in[/td][/tr][tr][td][math]3[/math][/td][td](you try)[/td][/tr][tr][td][math]4[/math][/td][td](you try)[/td][/tr][/table]

Suppose we have a 3-inch Jactus plant (on Day 0). [br]How tall is the Jactus on day 7? Be careful: Day 7 does not mean that [math]w=7[/math]. Instead, Day 7 corresponds to [math]w=1[/math]. Why? [br][br]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:[br]   a) How tall is a 3-inch Jactus at the start of Day 1? (Hint: What fraction of a week has elapsed?)[br]   b) How tall is the 3-inch Jactus on day 10? (Hint: What is [i]w[/i] now?) 

We will need to be able to describe how the height of a Jactus changes over time. [br][br]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: [br][list][*]We might say the height increased by 1 inch (this is an [b]additive relationship[/b]).[/*][*]Or we[b] [/b]might say the height increased by a [i]factor [/i]of 1.25 (a [b]multiplicative relationship[/b]). We are reporting the [b]scale factor[/b]. We can calculate it simply enough: scale factor = (new height) divided by (previous height).[/*][/list][br]Consider again the 3-inch Jactus plant from Question 2. [br](a) From an additive perspective, did the Jactus have a larger change in height on Day 0 or on Day 14?[br](b) And now from a multiplicative perspective: [br][list][*]What is the scale factor for the change in height on Day 0?[br][/*][*]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?[/*][/list]What do you notice? 

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