Use the image to answer the question.

Question

Four curves are graphed on a coordinate plane. The x-axis is labeled Time in Days and ranges from 0 to 6 in increments of 1. The y-axis is labeled Population and ranges from 0 to 25 in increments of 5.

The function f(x)=3(1.25)x
represents the population growth of a swarm of fruit flies. Create a graph modeling the equation, and determine which graph models the situation accurately.

(1 point)
Responses

Option #4
Option #4

Option #3
Option #3

Option #2
Option #2

Option #1
Option #1
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Answer 1

Based on the function \( f(x) = 3(1.25)^x \), we can analyze the characteristics of this exponential growth model.

1. Initial Value: At \( x = 0 \), \( f(0) = 3(1.25)^0 = 3(1) = 3 \). This means the population starts at 3.
2. Growth Rate: The base of the exponent (1.25) indicates that the population will increase by 25% each day.

Expected Characteristics of the Graph:

• The graph should start at the point (0, 3) on the y-axis.
• As \( x \) increases, the y-value should increase rapidly due to the nature of exponential growth.
• By Day 6, the population should be significantly higher than the starting point.

Estimating Population Growth:

• Day 1: \( f(1) = 3(1.25)^1 \approx 3.75 \)
• Day 2: \( f(2) = 3(1.25)^2 \approx 4.688 \)
• Day 3: \( f(3) = 3(1.25)^3 \approx 5.859 \)
• Day 4: \( f(4) = 3(1.25)^4 \approx 7.36 \)
• Day 5: \( f(5) = 3(1.25)^5 \approx 9.20 \)
• Day 6: \( f(6) = 3(1.25)^6 \approx 11.5 \)

Conclusion:

To identify which graph best represents this function, look for:

• A starting point at (0, 3).
• An upward curving trend that becomes steeper as time increases.
• The value should be around 3 at Day 0 and should continue to grow towards 11.5 at Day 6.

By evaluating the responses based on these characteristics, the correct option can be determined. Unfortunately, without seeing the images, I cannot specify which option is correct, but you can use this analysis to match with the options provided.