To find the equation of the exponential function represented by the table, let's analyze the pattern of values in the table.

For an exponential function, the general form is: $y = a \cdot b^x$ where:

• $a$ is the initial value (when $x = 0$),
• $b$ is the growth (or decay) factor.

Step 1: Identify $a$

From the table, when $x = 0$, $y = 1$. This means: $a = 1$ So, the function so far is: $y = 1 \cdot b^x = b^x$

Step 2: Identify $b$

To find $b$, we can use another point in the table. For example, when $x = 1$, $y = 4$. Substitute these values into the function to solve for $b$: $4 = b^1$ Thus, $b = 4$.

Step 3: Write the Equation

With $a = 1$ and $b = 4$, the exponential function is: $y = 4^x$

So, the answer is: $y = 4^x$

Would you like additional details on how this process works, or do you have any questions?

Here are some related questions to further explore this topic:

1. How would the equation change if the growth factor $b$ was a fraction?
2. What happens to the function if the initial value $a$ is not 1?
3. How can we identify exponential decay from a table?
4. Can this method be used for any set of exponential function values?
5. How can you verify that a given table represents an exponential function?

Tip: In an exponential function table, check the ratio between consecutive $y$-values to confirm that it is consistent; this suggests an exponential relationship.