When x= 0,1,2,3,4, y= 2,18,162,1458,13122 each. Determine this graph if it’s a linear or exponential, and also figure out a function of y=ax+b of this graph.

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.

When x = 0, 1, 2, 3, 4, y = 2, 18, 162, 1458, 13122 each. Determine if this graph is linear or exponential, and also figure out a function of y = ax + b for this graph.

Explore how to determine whether a set of points forms a linear or exponential graph. Analyze the given points (x = 0, 1, 2, 3, 4; y = 2, 18, 162, 1458, 13122), check for exponential growth, and derive the function y = 2 × 9^x. Suitable for Grades 9-12 students studying exponential functions.

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