**Exponential Model Derivation**To find the exponential model \( y = ae^{bx} \) that fits the points shown in the graph, we will analyze the graph in detail. The given points are:- \( (0, \frac{1}{2}) \)- \( (4, 6) \)**Graph Description:**The graph is a typical exponential curve starting from the point \( (0, \frac{1}{2}) \) on the y-axis and passing through \( (4, 6) \).**Steps to Find the Exponential Model:**1. **Substitute the Points into the Model:** - For \( x = 0 \), \( y = \frac{1}{2} \); this gives the equation: \[ \frac{1}{2} = ae^{b \cdot 0} = a \] Therefore, \( a = \frac{1}{2} \). - For \( x = 4 \), \( y = 6 \); substitute \( a = \frac{1}{2} \) and \( y = 6 \) into the equation: \[ 6 = \frac{1}{2}e^{4b} \]2. **Solve for \( b \):** \[ 6 \times 2 = e^{4b} \] \[ 12 = e^{4b} \] Take the natural logarithm on both sides: \[ \ln(12) = 4b \] \[ b = \frac{\ln(12)}{4} \] Calculate \( b \) rounded to four decimal places.3. **Rewrite the Model:**Substitute the values of \( a \) and \( b \) back into the exponential model to complete the formula.**Result:**\[ y = \frac{1}{2}e^{\left(\frac{\ln(12)}{4}\right)x} \]The exact expression for \( b \) is used to maintain precision. Ensure to calculate \( b \) to four decimal places to complete the answer.

Name: **Exponential Model Derivation**To find the exponential model \( y =
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Exponential Model Derivation**To find the exponential model \( y = ae^{bx} \) that fits the points shown in the graph, we will analyze the graph in detail. The given points are:- \( (0, \frac{1}{2}) \)- \( (4, 6) \)**Graph Description:**The graph