To find an exponential model for the dataset provided, we first need to analyze the y-values corresponding to the x-values.
From the table:
- When \( x = 0 \), \( y = 6 \).
- When \( x = 1 \), \( y = 12 \).
- When \( x = 2 \), \( y = 24 \).
- When \( x = 3 \), \( y = 48 \).
Let's look for a pattern in the y-values:
- The ratio of \( y \)-values when \( x \) increases by 1:
- From \( 0 \) to \( 1 \): \( \frac{12}{6} = 2 \)
- From \( 1 \) to \( 2 \): \( \frac{24}{12} = 2 \)
- From \( 2 \) to \( 3 \): \( \frac{48}{24} = 2 \)
This indicates that the y-values double as x increases by 1, suggesting an exponential growth with a base of 2.
The general form of an exponential function can be expressed as: \[ f(x) = a \cdot b^x \] where \( a \) is the initial value (y-intercept when \( x = 0 \)) and \( b \) is the growth factor.
From the table, when \( x = 0 \), \( y = 6 \), giving us \( a = 6 \) and since we found that \( b = 2 \), we can write the model as: \[ f(x) = 6 \cdot 2^x \]
Now, let's check our options:
- \( f(x) = 2(6)^x \)
- \( f(x) = 6(2^x) \)
- \( f(x) = 12^x \)
- \( f(x) = 6(2)^x \)
The correct exponential model for the data provided is: \[ f(x) = 6(2^x) \] Thus, the answer is: f(x) = 6(2^x)