Question
Identify the initial value and growth factor of each exponential function. Then determine if the function is exponential growth or decay
Answers
GPT-4o mini
To identify the initial value and growth factor of an exponential function, we usually express the function in the standard form:
\[ f(t) = a \cdot b^t \]
where:
- \( a \) is the initial value (the value of the function when \( t = 0 \))
- \( b \) is the growth factor
The function exhibits:
- Exponential growth if \( b > 1 \)
- Exponential decay if \( 0 < b < 1 \)
Let's look at some examples to clarify this:
1. **Function**: \( f(t) = 5 \cdot (2)^t \)
- Initial Value: \( a = 5 \)
- Growth Factor: \( b = 2 \) (since \( 2 > 1 \), it's exponential growth)
2. **Function**: \( f(t) = 3 \cdot (0.5)^t \)
- Initial Value: \( a = 3 \)
- Growth Factor: \( b = 0.5 \) (since \( 0 < 0.5 < 1 \), it's exponential decay)
3. **Function**: \( f(t) = 10 \cdot (1.1)^t \)
- Initial Value: \( a = 10 \)
- Growth Factor: \( b = 1.1 \) (since \( 1.1 > 1 \), it's exponential growth)
4. **Function**: \( f(t) = 7 \cdot (0.9)^t \)
- Initial Value: \( a = 7 \)
- Growth Factor: \( b = 0.9 \) (since \( 0 < 0.9 < 1 \), it's exponential decay)
If you have specific functions you would like me to analyze, please provide them, and I can help identify the initial values, growth factors, and whether they represent exponential growth or decay.
\[ f(t) = a \cdot b^t \]
where:
- \( a \) is the initial value (the value of the function when \( t = 0 \))
- \( b \) is the growth factor
The function exhibits:
- Exponential growth if \( b > 1 \)
- Exponential decay if \( 0 < b < 1 \)
Let's look at some examples to clarify this:
1. **Function**: \( f(t) = 5 \cdot (2)^t \)
- Initial Value: \( a = 5 \)
- Growth Factor: \( b = 2 \) (since \( 2 > 1 \), it's exponential growth)
2. **Function**: \( f(t) = 3 \cdot (0.5)^t \)
- Initial Value: \( a = 3 \)
- Growth Factor: \( b = 0.5 \) (since \( 0 < 0.5 < 1 \), it's exponential decay)
3. **Function**: \( f(t) = 10 \cdot (1.1)^t \)
- Initial Value: \( a = 10 \)
- Growth Factor: \( b = 1.1 \) (since \( 1.1 > 1 \), it's exponential growth)
4. **Function**: \( f(t) = 7 \cdot (0.9)^t \)
- Initial Value: \( a = 7 \)
- Growth Factor: \( b = 0.9 \) (since \( 0 < 0.9 < 1 \), it's exponential decay)
If you have specific functions you would like me to analyze, please provide them, and I can help identify the initial values, growth factors, and whether they represent exponential growth or decay.