The given data points are:

• $x = 0, 1, 2, 3, 4$
• $y = 2, 18, 162, 1458, 13122$

Step 1: Check for linearity

A linear function follows the form $y = ax + b$. To check for linearity, we calculate the differences between consecutive $y$-values. If these differences are constant, the function is linear.

1. $18 - 2 = 16$
2. $162 - 18 = 144$
3. $1458 - 162 = 1296$
4. $13122 - 1458 = 11664$

These differences are not constant, so the function is not linear.

Step 2: Check for exponential growth

An exponential function follows the form $y = ab^x$. In exponential functions, the ratio between consecutive $y$-values should be constant.

Let's check the ratios:

1. $\frac{18}{2} = 9$
2. $\frac{162}{18} = 9$
3. $\frac{1458}{162} = 9$
4. $\frac{13122}{1458} = 9$

Since the ratios are constant, the function is exponential.

Step 3: Determine the function

The general form of an exponential function is:

$y = ab^x$

To find the values of $a$ and $b$, we use the known points:

• When $x = 0$, $y = 2$: $y = ab^0 \implies y = a \implies a = 2$

Thus, the equation becomes:

$y = 2b^x$

• Use another point, say $x = 1$, $y = 18$, to solve for $b$: $18 = 2b^1 \implies b = \frac{18}{2} = 9$

So, the function is:

$y = 2 \times 9^x$

Final Answer:

The function describing the data is exponential, and the equation is:

$y = 2 \times 9^x$

Do you want more details or have any questions?

Here are five related questions:

1. How would you calculate the inverse of this exponential function?
2. What are the properties of exponential functions in general?
3. Can you determine the asymptotes for the function $y = 2 \times 9^x$?
4. How does the graph of this function behave as $x \to \infty$?
5. How would the graph change if the base $9$ were replaced with a different number?

Tip: Exponential functions grow or decay at rates proportional to their current value, unlike linear functions which grow at a constant rate.