Question

Use the image to answer the 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 #4
Option #4

Option #2
Option #2

Option #1

Answer 1

To determine which graph accurately models the function \( f(x) = 3(1.25)^x \), we first analyze the function.

This function represents exponential growth, starting at an initial population of 3 when \( x = 0 \). As \( x \) (Time in Days) increases, the function grows by a factor of 1.25 each day.

Now we can calculate some specific values:

- For \( x = 0 \):
\[
f(0) = 3(1.25)^0 = 3(1) = 3
\]

- For \( x = 1 \):
\[
f(1) = 3(1.25)^1 = 3(1.25) = 3.75
\]

- For \( x = 2 \):
\[
f(2) = 3(1.25)^2 = 3(1.5625) = 4.6875
\]

- For \( x = 3 \):
\[
f(3) = 3(1.25)^3 = 3(1.953125) \approx 5.859375
\]

- For \( x = 4 \):
\[
f(4) = 3(1.25)^4 = 3(2.44140625) \approx 7.32421875
\]

- For \( x = 5 \):
\[
f(5) = 3(1.25)^5 = 3(3.0517578125) \approx 9.1552734375
\]

- For \( x = 6 \):
\[
f(6) = 3(1.25)^6 = 3(3.814697265625) \approx 11.444091796875
\]

When visually inspecting the graphs labeled Option #1, Option #2, Option #3, and Option #4, we will look for a graph that starts at approximately 3 when \( x = 0 \) and rises slightly more than previous points as \( x \) increases (showing exponential growth).

If you provide me with the descriptions or properties of the curves present in the four options (like their starting points, slopes, shapes, etc.), I can help you choose the correct option among them.