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 #3
Option #3

Option #1
Option #1

Option #4
Option #4

Option #2

Answer 1

To determine which graph accurately models the function \( f(x) = 3(1.25)^x \), we need to analyze the characteristics of the function:

1. Initial Value: At \( x = 0 \), \[ f(0) = 3(1.25)^0 = 3 \] This indicates that the population starts at 3.

2. Growth Rate: The base of the exponent, 1.25, indicates that the population increases by 25% each day. The function will grow exponentially.

3. End Behavior: As \( x \) increases, the value of \( f(x) \) will increase exponentially but will do so starting from 3 on the y-axis.

When analyzing the four options (graphs), look for:

• A y-intercept at 3.
• A curve that rises slowly at first (since it's exponential growth starting from 3) and then accelerates upwards over time due to the growth factor of 1.25.

You should choose the graph that starts at (0, 3) and shows a steady, exponential increase as x moves from 0 to 6.

Without seeing the actual graphs, I recommend you choose the one that best matches this behavior, particularly focusing on the correct initial value and the general shape of exponential growth.