To determine whether the table represents a linear or nonlinear function, we can check the rate of change (also known as the slope) between consecutive pairs of points. A linear function will have a constant rate of change (slope), while a nonlinear function will not.

The points given are:

• (5, -9)
• (7, -3)
• (9, -1)
• (11, 3)

Step 1: Calculate the change in $y$ and $x$ between consecutive points.

For the first pair of points, (5, -9) and (7, -3):

• Change in $x$: $7 - 5 = 2$
• Change in $y$: $-3 - (-9) = -3 + 9 = 6$
• Slope: $\frac{6}{2} = 3$

For the second pair of points, (7, -3) and (9, -1):

• Change in $x$: $9 - 7 = 2$
• Change in $y$: $-1 - (-3) = -1 + 3 = 2$
• Slope: $\frac{2}{2} = 1$

For the third pair of points, (9, -1) and (11, 3):

• Change in $x$: $11 - 9 = 2$
• Change in $y$: $3 - (-1) = 3 + 1 = 4$
• Slope: $\frac{4}{2} = 2$

Step 2: Check if the slope is constant.

The slopes calculated are:

• $3$ between the first pair,
• $1$ between the second pair, and
• $2$ between the third pair.

Since the slopes are not consistent, the rate of change is not constant. Therefore, the function is nonlinear.

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Do you want more details on how the slope works, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you find the slope of a line between two points?
2. What is the geometric interpretation of a linear function?
3. How can you use a table to determine if a function is linear or nonlinear?
4. What are some characteristics of nonlinear functions?
5. Can a function be linear over a certain interval and nonlinear elsewhere?

Tip: If you have a table of values, always check the consistency of the slope to determine if the function is linear!