To create an exponential model based on the given dataset, we can first analyze the y-values to find a pattern. Observing the y-values:
- For \( x = 0 \), \( y = 6 \)
- For \( x = 1 \), \( y = 12 \)
- For \( x = 2 \), \( y = 24 \)
- For \( x = 3 \), \( y = 48 \)
Let's look at the ratios between successive y-values:
- From \( 6 \) to \( 12 \) is a factor of \( 2 \) (12/6 = 2)
- From \( 12 \) to \( 24 \) is also a factor of \( 2 \) (24/12 = 2)
- From \( 24 \) to \( 48 \) is again a factor of \( 2 \) (48/24 = 2)
This consistent doubling suggests an exponential model of the form:
\[ f(x) = a \cdot b^x \]
We know \( f(0) = 6 \). Therefore:
\[ 6 = a \cdot b^0 \Rightarrow a = 6 \]
Now, knowing the ratio motif (b = 2), we can express the model as follows:
\[ f(x) = 6 \cdot 2^x \]
Thus, the appropriate exponential model that fits the data is:
\[ f(x) = 6(2^x) \]
So the correct response is:
f(x) = 6(2^x).