Question
an insect population after x months can be modeled by the graph at left. complete the table below and write the exponential function that describes this scenario.
months, x: 0,1,2,3
population, y: __,__,__,__
y=
months, x: 0,1,2,3
population, y: __,__,__,__
y=
Answers
GPT-4o mini
To provide you with a complete response, I would typically need information about the graph's data to fill in the insect population values for the given months. However, since I can't see the graph, I'll guide you through the process of how to derive the values and the exponential function.
### Steps to Complete the Table:
1. **Read the Graph:** Look at the graph and note the population values for each month (0, 1, 2, 3). Record these values for y.
2. **Identify a Pattern:** If the population seems to grow at a consistent factor (for example, doubling every month), that indicates exponential growth.
3. **Exponential Function Form:** The standard form for an exponential function describing population growth is:
\[
y = a \cdot b^x
\]
where:
- \(y\) is the population,
- \(a\) is the initial population (when \(x = 0\)),
- \(b\) is the growth factor,
- \(x\) is the number of months.
4. **Determine \(a\) and \(b\):** The initial population \(a\) can be found directly from the first entry in the table. The growth factor \(b\) can be calculated by taking the ratio of the population at consecutive months.
### Example:
Suppose the graph shows the following values:
- At month \(0\) (x=0), the population is \(100\).
- At month \(1\) (x=1), the population is \(150\).
- At month \(2\) (x=2), the population is \(225\).
- At month \(3\) (x=3), the population is \(337.5\).
Then the table would look like:
- Months, \(x\): \(0, 1, 2, 3\)
- Population, \(y\): \(100, 150, 225, 337.5\)
To find the exponential function:
1. \( a = 100 \) (the population at \(0\) months).
2. To find \( b \):
\[
b = \frac{y(1)}{y(0)} = \frac{150}{100} = 1.5
\]
(You can verify that the growth factor remains consistent for the other months).
Therefore, the exponential function that models this population growth would be:
\[
y = 100 \cdot (1.5)^x
\]
### Final Output:
Fill the table, and you would get:
- Months, \(x\): \(0, 1, 2, 3\)
- Population, \(y\): populate according to the values retrieved from the graph.
- The exponential function \(y = 100 \cdot (1.5)^x\) (just as an example).
If you can provide the population values from your graph, I can help you further refine this answer!
### Steps to Complete the Table:
1. **Read the Graph:** Look at the graph and note the population values for each month (0, 1, 2, 3). Record these values for y.
2. **Identify a Pattern:** If the population seems to grow at a consistent factor (for example, doubling every month), that indicates exponential growth.
3. **Exponential Function Form:** The standard form for an exponential function describing population growth is:
\[
y = a \cdot b^x
\]
where:
- \(y\) is the population,
- \(a\) is the initial population (when \(x = 0\)),
- \(b\) is the growth factor,
- \(x\) is the number of months.
4. **Determine \(a\) and \(b\):** The initial population \(a\) can be found directly from the first entry in the table. The growth factor \(b\) can be calculated by taking the ratio of the population at consecutive months.
### Example:
Suppose the graph shows the following values:
- At month \(0\) (x=0), the population is \(100\).
- At month \(1\) (x=1), the population is \(150\).
- At month \(2\) (x=2), the population is \(225\).
- At month \(3\) (x=3), the population is \(337.5\).
Then the table would look like:
- Months, \(x\): \(0, 1, 2, 3\)
- Population, \(y\): \(100, 150, 225, 337.5\)
To find the exponential function:
1. \( a = 100 \) (the population at \(0\) months).
2. To find \( b \):
\[
b = \frac{y(1)}{y(0)} = \frac{150}{100} = 1.5
\]
(You can verify that the growth factor remains consistent for the other months).
Therefore, the exponential function that models this population growth would be:
\[
y = 100 \cdot (1.5)^x
\]
### Final Output:
Fill the table, and you would get:
- Months, \(x\): \(0, 1, 2, 3\)
- Population, \(y\): populate according to the values retrieved from the graph.
- The exponential function \(y = 100 \cdot (1.5)^x\) (just as an example).
If you can provide the population values from your graph, I can help you further refine this answer!