The exponential function  f(x)=abx
  models the volume of water in an inverted cone after x seconds of draining. As x increases, the instantaneous rate of change gets closer to 0.  

What is a possible growth factor for  f(x)
 ?


One possible growth factor is 1.  


One possible growth factor is –1.56.  


One possible growth factor is 0.33.  


One possible growth factor is 1.36.  

For the given exponential function \( f(x) = ab^x \), where \( a \) is a constant and \( b \) is the base (growth factor), the behavior of the function depends on the value of \( b \). 

Given that the function models the volume of water draining from an inverted cone, and the rate of change approaches 0 as \( x \) increases, this suggests that the volume is decreasing over time.

Now, let's examine the possible growth factors:

- **1**: A growth factor of 1 means no change in the volume over time, which doesn't fit the description of water draining from the cone.
- **–1.56**: A negative growth factor is unrealistic in this context because it would imply alternating values (positive and negative), which is not possible for volume.
- **0.33**: A growth factor between 0 and 1 indicates exponential decay, which aligns with the scenario of water draining, as the volume decreases over time.
- **1.36**: A growth factor greater than 1 would imply exponential growth, which is not appropriate for this situation since the water volume is decreasing.

Thus, the **possible growth factor is 0.33** because it represents exponential decay, fitting the situation of water draining from the cone.

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. How does the initial value \( a \) influence the function \( f(x) \)?
2. What is the meaning of the growth factor in the context of exponential functions?
3. How would the function behave if the growth factor was exactly 1?
4. What happens to the function \( f(x) = ab^x \) when \( b > 1 \)?
5. Can the growth factor in an exponential function be negative? Why or why not?

**Tip:** Exponential decay occurs when the growth factor \( b \) is between 0 and 1, which results in the function decreasing over time.

The exponential function f(x) = ab^x models the volume of water in an inverted cone after x seconds of draining. As x increases, the instantaneous rate of change gets closer to 0. What is a possible growth factor for f(x)?

Explore how the exponential function f(x) = ab^x models the volume of water draining from an inverted cone. Understand why a growth factor of 0.33 represents exponential decay in this scenario, as the volume decreases over time. Suitable for Grades 9-12, this explanation covers key concepts in exponential decay and the behavior of the function.

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