To create an exponential model from the given data points, we need to formulate the equations based on the typical form of an exponential function, which is:
\[ y = a \cdot b^x \]
Where:
- \( y \) is the dependent variable,
- \( x \) is the independent variable,
- \( a \) is the initial value (the value of \( y \) when \( x = 0 \)),
- \( b \) is the growth factor.
Based on the provided data:
- \( (x, y) = (-1, 3) \)
- \( (0, 6) \)
- \( (1, 12) \)
- \( (2, 24) \)
- \( (3, 48) \)
From the point where \( x = 0 \) and \( y = 6 \), we can deduce that \( a = 6 \), so the equation can be rewritten as:
\[ y = 6 \cdot b^x \]
Next, we can use the other points to solve for \( b \).
Using the point \( (1, 12) \): \[ 12 = 6 \cdot b^1 \] \[ b = \frac{12}{6} = 2 \]
Using the point \( (2, 24) \) to verify: \[ 24 = 6 \cdot b^2 \] Substituting \( b = 2 \): \[ 24 = 6 \cdot 2^2 \] \[ 24 = 6 \cdot 4 = 24 \] (This verifies.)
Using the point \( (3, 48) \) to validate: \[ 48 = 6 \cdot b^3 \] Substituting again: \[ 48 = 6 \cdot 2^3 \] \[ 48 = 6 \cdot 8 = 48 \] (This also verifies.)
Thus, the exponential model that fits the data set is:
\[ y = 6 \cdot 2^x \]
This is the equation that models the data points provided.