To find the equation of the exponential function represented by the table, we can assume that the function has the form:

$y = a \cdot b^x$

where $a$ is the initial value (when $x = 0$) and $b$ is the base of the exponential function (growth factor).

Step 1: Identify the value of $a$:

From the table, when $x = 0$, $y = 2$. So, we have:

$a = 2$

Step 2: Find the base $b$:

We can use another point from the table to find the value of $b$. Let's use the point $(1, 6)$. Substitute $x = 1$, $y = 6$, and $a = 2$ into the exponential equation:

$6 = 2 \cdot b^1$

$b = \frac{6}{2} = 3$

Step 3: Write the equation:

Now we know both $a$ and $b$, so we can write the equation of the exponential function:

$y = 2 \cdot 3^x$

Step 4: Verify with the other points:

• For $x = 2$:
  $y = 2 \cdot 3^2 = 2 \cdot 9 = 18 \quad (\text{which matches the table})$

• For $x = 3$:
  $y = 2 \cdot 3^3 = 2 \cdot 27 = 54 \quad (\text{which also matches the table})$

Hence, the exponential function that fits the table is:

$\boxed{y = 2 \cdot 3^x}$

Would you like more details on this solution, or do you have any questions?

Relative Questions to Consider:

1. How would the equation change if the base $b$ was a fraction instead of a whole number?
2. What is the impact of the coefficient $a$ on the shape of the exponential graph?
3. Can we determine the equation of an exponential function using only two points?
4. What would happen to the function if the base $b$ was negative?
5. How do transformations (such as translations or reflections) affect the equation of the exponential function?

Tip:

When given a table of values for an exponential function, always look for a common ratio between consecutive $y$-values to quickly identify the base of the function.