Answer:
1. The exponential function that represents the spread of Ben's social media post is f(x) = 2 * 3^x. This function shows that the number of shares starts at 2 and triples every day.
2. The exponential function that represents the spread of Carter's social media post is f(x) = 10 * 2^x. This function shows that the number of shares starts at 10 and doubles every day.
3. To graph these functions, we can choose three points for each curve and plot them on the same coordinate plane. For Ben's function (f(x) = 2 * 3^x), we can use the points (0, 2), (1, 6), and (2, 18). For Carter's function (f(x) = 10 * 2^x), we can use the points (0, 10), (1, 20), and (2, 40). By connecting these points, we can visualize the growth of shares over time.
4. To predict the number of shares on Day 3 and Day 10 for each student, we can substitute the values of x into their respective exponential functions.
- For Ben's social media post (f(x) = 2 * 3^x):
- On Day 3: f(3) = 2 * 3^3 = 54 shares
- On Day 10: f(10) = 2 * 3^10 = 1,458 shares
- For Carter's social media post (f(x) = 10 * 2^x):
- On Day 3: f(3) = 10 * 2^3 = 80 shares
- On Day 10: f(10) = 10 * 2^10 = 1,280 shares
These predictions are based on the patterns observed in the given table and the exponential functions we derived.
5. If Amber decides to mail copies of her photo to the 45 residents of her grandmother's assisted living facility, the new function representing her photo shares would be f(x) = 3(4) + 45. This new function takes into account the initial 45 shares from mailing the copies, in addition to the existing growth pattern.
Comparing this new graph with the original graph of Amber's photo shares, we would see a parallel shift upwards by 45 units on the y-axis. The rate of growth would still follow the same pattern, but the initial number of shares would be higher.
6. Based on the results, Ben's post travels the fastest. This is shown in the equation form of the functions by looking at the base of the exponent. In Ben's function (f(x) = 2 * 3^x), the base is 3, indicating exponential growth at a faster rate compared to Carter's function (f(x) = 10 * 2^x), where the base is 2. A higher base leads to faster growth. Therefore, Ben's post spreads faster than Carter's post.
Explanation: