To determine if the function represented by this table is linear or nonlinear, we can examine the change in $y$ values as $x$ increases. For a function to be linear, the change in $y$ should be constant when $x$ changes by a fixed amount.

The table shows:

Step 1: Calculate the Change in $y$ as $x$ Changes

1. From $x = 3$ to $x = 4$:
   • Change in $y$ = $9 - 15 = -6$
2. From $x = 4$ to $x = 5$:
   • Change in $y$ = $2 - 9 = -7$

Step 2: Analyze the Pattern

The change in $y$ values is not consistent:

• First change in $y$ is $-6$.
• Second change in $y$ is $-7$.

Since the rate of change of $y$ with respect to $x$ is not constant, this indicates that the function is nonlinear.

Would you like a deeper explanation of linear and nonlinear functions or help with further questions?

Here are some related questions:

1. How do we determine the rate of change in a linear function?
2. Can we find a formula or equation that describes this nonlinear relationship?
3. What are other methods to identify if a function is linear or nonlinear?
4. How would this table look if the function were linear?
5. What are some real-world examples of nonlinear functions?

Tip: In a linear function, the difference in $y$ values for equal increments in $x$ should always be the same, indicating a constant slope.