Question
This question has four parts.
Martha rescued some bunnies caught in thorns in her yard. She set up a home for them on her back patio.
Before long, the bunnies began having babies. She recorded the number of bunnies each month in this table.
table:
Months 0 1 2 3
Bunnies 5 15 45 135
Part A
Classify the function.
(1 point)
Responses
linear
linear
quadratic
quadratic
cubic
cubic
exponential
exponential
Question 2
Part B
Determine the initial number of bunnies.
(1 point)
Question 3
Part C
Write a function in the form of f(x)=a⋅(b)^x
that gives f(x),
the number of bunnies, after x
months.
(1 point)
f(x)=
Question 4
Part D
Find the average rate of change in the number of bunnies per month between months 1 and 3.
(1 point)
Responses
30
30
60
60
90
90
120
Martha rescued some bunnies caught in thorns in her yard. She set up a home for them on her back patio.
Before long, the bunnies began having babies. She recorded the number of bunnies each month in this table.
table:
Months 0 1 2 3
Bunnies 5 15 45 135
Part A
Classify the function.
(1 point)
Responses
linear
linear
quadratic
quadratic
cubic
cubic
exponential
exponential
Question 2
Part B
Determine the initial number of bunnies.
(1 point)
Question 3
Part C
Write a function in the form of f(x)=a⋅(b)^x
that gives f(x),
the number of bunnies, after x
months.
(1 point)
f(x)=
Question 4
Part D
Find the average rate of change in the number of bunnies per month between months 1 and 3.
(1 point)
Responses
30
30
60
60
90
90
120
Answers
GPT-4o mini
Let’s address each part of the question systematically.
### Part A: Classify the function.
To determine the classification of the function, we can analyze the number of bunnies each month:
- Month 0: 5 bunnies
- Month 1: 15 bunnies
- Month 2: 45 bunnies
- Month 3: 135 bunnies
If we look at the ratios:
- \( \frac{15}{5} = 3 \)
- \( \frac{45}{15} = 3 \)
- \( \frac{135}{45} = 3 \)
The number of bunnies triples each month, suggesting the growth is exponential rather than linear, quadratic, or cubic.
**Answer: Exponential**
---
### Part B: Determine the initial number of bunnies.
The initial number of bunnies corresponds to the number at month 0.
**Answer: 5 bunnies**
---
### Part C: Write a function in the form of \( f(x)=a⋅(b)^x \).
From our previous analysis:
- \( a \) which represents the initial number of bunnies is 5.
- The growth factor \( b \) is 3 (since the bunnies triple each month).
Thus, the function can be written as:
\[ f(x) = 5 \cdot (3)^x \]
**Answer: \( f(x)=5 \cdot (3)^x \)**
---
### Part D: Find the average rate of change in the number of bunnies per month between months 1 and 3.
To calculate the average rate of change between two points, we use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Where \( a = 1 \) and \( b = 3 \):
- \( f(1) = 15 \) (the number of bunnies at month 1)
- \( f(3) = 135 \) (the number of bunnies at month 3)
Plug these values into the formula:
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{135 - 15}{2} = \frac{120}{2} = 60
\]
**Answer: 60**
---
In summary:
- Part A: **Exponential**
- Part B: **5 bunnies**
- Part C: **\( f(x)=5 \cdot (3)^x \)**
- Part D: **60**
### Part A: Classify the function.
To determine the classification of the function, we can analyze the number of bunnies each month:
- Month 0: 5 bunnies
- Month 1: 15 bunnies
- Month 2: 45 bunnies
- Month 3: 135 bunnies
If we look at the ratios:
- \( \frac{15}{5} = 3 \)
- \( \frac{45}{15} = 3 \)
- \( \frac{135}{45} = 3 \)
The number of bunnies triples each month, suggesting the growth is exponential rather than linear, quadratic, or cubic.
**Answer: Exponential**
---
### Part B: Determine the initial number of bunnies.
The initial number of bunnies corresponds to the number at month 0.
**Answer: 5 bunnies**
---
### Part C: Write a function in the form of \( f(x)=a⋅(b)^x \).
From our previous analysis:
- \( a \) which represents the initial number of bunnies is 5.
- The growth factor \( b \) is 3 (since the bunnies triple each month).
Thus, the function can be written as:
\[ f(x) = 5 \cdot (3)^x \]
**Answer: \( f(x)=5 \cdot (3)^x \)**
---
### Part D: Find the average rate of change in the number of bunnies per month between months 1 and 3.
To calculate the average rate of change between two points, we use the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Where \( a = 1 \) and \( b = 3 \):
- \( f(1) = 15 \) (the number of bunnies at month 1)
- \( f(3) = 135 \) (the number of bunnies at month 3)
Plug these values into the formula:
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{135 - 15}{2} = \frac{120}{2} = 60
\]
**Answer: 60**
---
In summary:
- Part A: **Exponential**
- Part B: **5 bunnies**
- Part C: **\( f(x)=5 \cdot (3)^x \)**
- Part D: **60**