To find the equation of the exponential function represented by the table, we observe that the table shows values of $y$ that change exponentially as $x$ increases.

1. Identify the form of the exponential equation: The general form of an exponential function is: $y = a \cdot b^x$ where $a$ is the initial value (when $x = 0$) and $b$ is the growth factor.

2. Find the value of $a$: From the table, when $x = 0$, $y = 4$. So, $a = 4$.

3. Determine the value of $b$: We can find $b$ by looking at the ratio of successive $y$-values:

   • From $x = 0$ to $x = 1$: $\frac{y(1)}{y(0)} = \frac{8}{4} = 2$
   • From $x = 1$ to $x = 2$: $\frac{y(2)}{y(1)} = \frac{16}{8} = 2$
   • From $x = 2$ to $x = 3$: $\frac{y(3)}{y(2)} = \frac{32}{16} = 2$

   This confirms that $b = 2$, as the ratio is consistent across each step.

4. Write the equation: Substituting $a = 4$ and $b = 2$, we get: $y = 4 \cdot 2^x$

Final Answer

The equation of the exponential function is: $y = 4 \cdot 2^x$

Would you like further details or have any questions?

Here are some related questions for further practice:

1. How would you determine the equation if the values in the table were decreasing instead of increasing?
2. What if the initial value $y(0)$ was different, say $y(0) = 5$, but the growth factor was still 2?
3. How can you use logarithms to solve for unknown variables in exponential functions?
4. What are the differences between exponential growth and decay functions?
5. How can you find the rate of change in exponential functions?

Tip: Always check the ratio between $y$-values in exponential tables to confirm the growth factor $b$.