To determine the exponential growth equation in the form \( y = a(b)^x \), we first need to identify the initial value and the common ratio from the given data points.
The points given are:
- \( (0, 4) \)
- \( (1, 7.6) \)
- \( (2, 14.44) \)
Step 1: Find the Initial Value
The initial value of the exponential equation occurs when \( x = 0 \). According to the first point \( (0, 4) \):
- The initial value \( a \) is: \[ a = 4 \]
Step 2: Calculate the Common Ratio
The common ratio \( b \) can be determined by analyzing the growth from one point to the next. To find \( b \), we can use the values from the first two points:
From \( (0, 4) \) to \( (1, 7.6) \): \[ b = \frac{y(1)}{y(0)} = \frac{7.6}{4} = 1.9 \]
You can verify this by checking the growth from \( (1, 7.6) \) to \( (2, 14.44) \): \[ b = \frac{y(2)}{y(1)} = \frac{14.44}{7.6} \approx 1.9 \]
Final Equation
Using \( a \) and \( b \): \[ y = 4(1.9)^x \]
Thus, we can fill in:
- \( y = 4(1.9)^x \)
- The initial value is \( 4 \).
- The common ratio is \( 1.9 \).
Summary
- Equation: \( y = 4(1.9)^x \)
- Initial Value: \( 4 \)
- Common Ratio: \( 1.9 \)
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