Which is the initial value that shrinks an exponential growth function by \( 50 \% \) ?

To determine the initial value that causes an exponential growth function to **shrink by 50%**, it's helpful to clarify the terminology and the behavior of exponential functions. ### Understanding Exponential Functions An **exponential function** generally has the form: \[ f(t) = A \cdot e^{kt} \] - **\( A \)**: Initial value (the value of the function at time \( t = 0 \)) - **\( k \)**: Growth rate constant - If \( k > 0 \), the function **grows exponentially**. - If \( k < 0 \), the function **decays exponentially**. Given that exponential **growth** implies the function increases over time, to have the function **shrink by 50%**, we're essentially looking at exponential **decay**. This means we need to adjust the initial value \( A \) such that the function's value decreases by half over a specified period. ### Calculating the Initial Value for a 50% Shrink Assuming you want the function to **decay to 50% of its initial value** at a specific time \( t \), you can set up the equation: \[ \frac{f(t)}{A} = 0.5 \] Substituting the exponential function: \[ \frac{A \cdot e^{kt}}{A} = 0.5 \] \[ e^{kt} = 0.5 \] Taking the natural logarithm of both sides: \[ kt = \ln(0.5) \] \[ k = \frac{\ln(0.5)}{t} \] However, **if the goal is simply to set the initial value so that the entire function is scaled down by 50% from the outset**, you would adjust the initial value directly: \[ A_{\text{new}} = 0.5 \cdot A_{\text{original}} \] ### Conclusion To **shrink an exponential function by 50%**, you should set the **initial value to half** of its original value. This effectively scales the entire function down, ensuring that all future values of the function are 50% of what they would have been with the original initial value. **Example:** - **Original Function:** \( f(t) = A \cdot e^{kt} \) - **Scaled-Down Function:** \( f_{\text{new}}(t) = 0.5A \cdot e^{kt} \) This adjustment ensures that the function starts at half its original value and maintains the same growth or decay rate. **Summary:** To shrink an exponential function by 50%, set its initial value to half of the original initial value.