The table provided shows the relationship between the variables $x$ and $y$. It seems you want to find an exponential function of the form $y = ab^x$ that fits this data.

To determine the exponential function, we can use the given data points to find the values of $a$ and $b$.

Step-by-Step Solution

1. Formulate the exponential equation: $y = ab^x$

2. Use the given points to create equations: For $x = 0$ and $y = 4$: $4 = ab^0$ $4 = a$ So, $a = 4$.

   Now we know the equation is: $y = 4b^x$

3. Find $b$ using another point: For $x = 1$ and $y = \frac{22}{5}$: $\frac{22}{5} = 4b^1$ $b = \frac{22}{5 \times 4}$ $b = \frac{22}{20}$ $b = 1.1$

4. Verify $b$ with other points: Let's use $x = 2$ and $y = \frac{121}{25}$: $\frac{121}{25} = 4(1.1)^2$ $\frac{121}{25} = 4 \times 1.21$ $\frac{121}{25} = 4.84$ $4.84 = 4.84$ This confirms our value of $b = 1.1$.

Thus, the exponential function that fits the data is: $y = 4 \cdot (1.1)^x$

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How would the function change if the base $b$ were a different value?
2. What is the significance of the initial value $a$ in an exponential function?
3. Can you graph the function $y = 4 \cdot (1.1)^x$ and describe its behavior?
4. How would you determine the exponential function if the data points were not exact powers of a single base?
5. How does the exponential function $y = 4 \cdot (1.1)^x$ behave as $x$ approaches infinity and negative infinity?

Tip: When verifying an exponential function, always check multiple data points to ensure the accuracy of your $a$ and $b$ values.