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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
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

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.

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.

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.

1 answer

To fill out the population chart of beetles using the equation \( f(x) = 350(1.22)^x \), we will calculate the number of beetles for each month \( x \) from 0 to 5.

  1. For \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350(1) = 350 \]

  2. For \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350(1.22) \approx 427 \]

  3. For \( x = 2 \): \[ f(2) = 350(1.22)^2 = 350(1.4884) \approx 521 \]

  4. For \( x = 3 \): \[ f(3) = 350(1.22)^3 = 350(1.8144) \approx 635 \]

  5. For \( x = 4 \): \[ f(4) = 350(1.22)^4 = 350(2.2198) \approx 777 \]

  6. For \( x = 5 \): \[ f(5) = 350(1.22)^5 = 350(2.7081) \approx 947 \]

Now, let's summarize the complete data in a chart format:

| Month (x) | Beetles (f(x)) | |-----------|-----------------| | 0 | 350 | | 1 | 427 | | 2 | 521 | | 3 | 635 | | 4 | 777 | | 5 | 947 |

Plot Points

The points to plot are:

  • (0, 350)
  • (1, 427)
  • (2, 521)
  • (3, 635)
  • (4, 777)
  • (5, 947)

Graph Selection

The growth rate of 22% per month means that the graph of the function will be an increasing curve, not a decreasing one. Therefore, the correct graph to choose must show an increasing trend that best matches the calculated points.

Given the descriptions of the images you provided:

  1. A decreasing curve with an arrow at the end.
  2. An increasing curve with an arrow on the top.
  3. A decreasing curve with an arrow at the end.

The correct response would be the second graph (the increasing curve), as it reflects the nature of exponential growth in the population of beetles.

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