To find an exponential equation that fits the data given in the table, we can look for a general form of an exponential equation, which is:
\[ y = a \cdot b^x \]
where \( a \) and \( b \) are constants.
From the values in the table:
| x | y | |---|----| | 1 | 12 | | 2 | 36 | | 3 | 108 | | 4 | 324 |
Let's determine the constants \( a \) and \( b \):
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Start with the point when \( x = 1 \): \[ y = a \cdot b^1 = 12 \] Thus, we have: \[ a \cdot b = 12 \quad (1) \]
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For \( x = 2 \): \[ y = a \cdot b^2 = 36 \] Thus, we have: \[ a \cdot b^2 = 36 \quad (2) \]
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Dividing equation (2) by equation (1): \[ \frac{a \cdot b^2}{a \cdot b} = \frac{36}{12} \] \[ b = 3 \]
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Now substitute \( b = 3 \) back into equation (1): \[ a \cdot 3 = 12 \] \[ a = 4 \]
Now we have both values:
- \( a = 4 \)
- \( b = 3 \)
Putting this together, the exponential equation is:
\[ \boxed{y = 4 \cdot 3^x} \]
This is the exponential function that corresponds to the data points in your table.