Question

Exponential Functions from Situations Quick Check
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Question
Use the table to answer the question.

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A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)x can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)
Responses

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.

An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.

An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 2,000 in increments of 200. The y-axis for Beetles ranges from 0 to 10 in increments of 1.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 2,000 in increments of 200. The y-axis for Beetles ranges from 0 to 10 in increments of 1.

A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
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Answer 1

To complete the table based on the given exponential growth model \( f(x) = 350(1.22)^x \) for a population of beetles increasing at a rate of 22% per month, we'll calculate the number of beetles at each specified month:

1. **Month 0**:
\[
f(0) = 350(1.22)^0 = 350 \times 1 = 350
\]
2. **Month 1**:
\[
f(1) = 350(1.22)^1 = 350 \times 1.22 \approx 427
\]
3. **Month 2**:
\[
f(2) = 350(1.22)^2 = 350 \times 1.4884 \approx 520
\]
4. **Month 3**:
\[
f(3) = 350(1.22)^3 = 350 \times 1.8038 \approx 631
\]
5. **Month 4**:
\[
f(4) = 350(1.22)^4 = 350 \times 2.198 \approx 769
\]
6. **Month 5**:
\[
f(5) = 350(1.22)^5 = 350 \times 2.683 \approx 939
\]

Now, let's summarize the completed table:

| Month (x) | Population (f(x)) |
|-----------|--------------------|
| 0 | 350 |
| 1 | 427 |
| 2 | 520 |
| 3 | 631 |
| 4 | 769 |
| 5 | 939 |

### Conclusion
The function models exponential growth, so the graph will be an increasing curve as the population of beetles increases over time.

Therefore, the correct graph should show an **increasing curve** with an arrow on the top. The response that fits this description is:

**An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.**

This graph accurately represents the growth of the beetle population over time based on the provided model.