To find the exponential function that fits the given points \( (0, 2) \), \( (1, 6) \), and \( (2, 18) \), we can assume the form of the function is:
\[ y = a \cdot b^x \]
where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base of the exponential function.
From the first point \( (0, 2) \), we can determine that:
\[ y = a \cdot b^0 = a \cdot 1 = a \Rightarrow a = 2 \]
Now, we can substitute \( a \) into the function to get:
\[ y = 2 \cdot b^x \]
Next, we'll use the second point \( (1, 6) \):
\[ 6 = 2 \cdot b^1 \]
This simplifies to:
\[ b = \frac{6}{2} = 3 \]
Now we can substitute \( b \) back into our function:
\[ y = 2 \cdot 3^x \]
We'll verify this function with the third point \( (2, 18) \):
\[ y = 2 \cdot 3^2 = 2 \cdot 9 = 18 \]
This is correct. Therefore, the exponential function that fits the given points is:
\[ y = 2 \cdot 3^x \]