Question

COMPARING THE GROWTH OF BLOGS

Jayden, Keiko, and Lauren each start writing a blog at the same time.

When Jayden starts his blog (month 0), he gets 48 of his friends to
subscribe. At the end of each new month, he records the number of
subscribers in a table. His data for the first few months are shown in
the table to the right.
Jayden's Blog

Keiko finds that the number of subscribers K to her blog can be modeled by the following function rule, where m is the number of months since she started the blog:
K = M² + 10

Lauren was the only subscriber to her blog when she started (month 0). However, she found that her
number of subscribers doubled every month.

Your Task :
Determine and justify whether Jayden's, Keiko's or Lauren's blog will reach 200 followers first
using multiple representations. (Assume the subscribers continue to follow the established
pattern.)

Determine a realistic domain and range for this context and explain how you know it is reasonable.

Determine which representation of the functions is most effective to justify your claim and explain why.

Answer 1

To determine which blog will reach 200 followers first, we analyze Keiko's quadratic growth model and Lauren's exponential growth model. Lauren's blog will reach 200 subscribers in 8 months, faster than Keiko's. Graphs are the most effective representation to visually compare the growth and determine when each blog reaches the 200-subscriber milestone.

Step-by-step explanation:

To determine whether Jayden's, Keiko's, or Lauren's blog will reach 200 followers first, we need to compare the growth patterns of their respective subscriber bases over time. Jayden's growth is not provided in a specific function, but let's assume it follows a certain linear or polynomial trend. Keiko's subscribers follow the quadratic model K = m² + 10, where m is the number of months since the blog started. Lauren experiences exponential growth since her subscriber count doubles every month, which can be modeled by L = 2^m, starting with 1 subscriber at month 0.

For Keiko, to reach 200 subscribers, we solve the equation m² + 10 = 200 to get m² = 190. The smallest integer m that satisfies this is 14 (since 13.78 is approximately the square root of 190, but we need a complete month). For Lauren to reach 200 subscribers, we find the smallest m such that 2^m ≥ 200. This occurs when m = 8 (since 2^7=128 and 2^8=256).

Given these models, Lauren's blog would reach 200 subscribers first, followed by Keiko's and possibly Jayden's depending on his initial rate of growth, which we don't have enough information to model accurately. The most effective representation to justify this claim is through the use of graphs, as they can visually display the rapid growth of Lauren's subscriber count compared to the other two and clearly indicate when the 200-subscriber mark is reached by each blog.

A realistic domain and range for this context would be integer values of m (months) where m ≥ 0, as you cannot have a negative month count, and integer values of subscribers K and L where K, L ≥ 0 as you cannot have a negative number of subscribers.