To create an exponential equation from the given table, we look to express \( y \) in terms of \( x \).
Observing the pattern in the \( y \) values:
- When \( x = 1 \), \( y = 12 \)
- When \( x = 2 \), \( y = 36 \)
- When \( x = 3 \), \( y = 108 \)
- When \( x = 4 \), \( y = 324 \)
We can see that each \( y \) value appears to be a multiple of the previous one. Examining the ratios:
- \( \frac{36}{12} = 3 \)
- \( \frac{108}{36} = 3 \)
- \( \frac{324}{108} = 3 \)
This indicates that the pattern could be expressed as \( y = a \cdot b^x \), where \( a \) is a constant and \( b \) is the base indicating the growth factor determined from the ratios.
Since the growth factor is 3, we will set \( b = 3 \). Now we can find the value of \( a \) using one of the points. Using the point (1, 12):
\[ 12 = a \cdot 3^1 \]
This simplifies to:
\[ 12 = 3a \]
Solving for \( a \):
\[ a = \frac{12}{3} = 4 \]
Thus, we can write the exponential equation as:
\[ y = 4 \cdot 3^x \]
In conclusion, the exponential equation that fits the given table is:
\[ y = 4 \cdot 3^x \]
1 answer