Determine whether the given function is linear or nonlinear.  If it is​ linear, determine the slope.
x
y equals f left parenthesis x right parenthesis
0
negative 6
1
negative 2
2
0
3
1
4
0

To determine whether a function is linear, we need to check whether the change in \( y \) values (or \( f(x) \)) is consistent with a constant rate of change, or slope, as \( x \) increases by a constant amount.

### Step 1: Calculate the differences in \( x \) and \( y \).
- When \( x = 0 \), \( y = -6 \)
- When \( x = 1 \), \( y = -2 \)
- When \( x = 2 \), \( y = 0 \)
- When \( x = 3 \), \( y = 1 \)
- When \( x = 4 \), \( y = 0 \)

Now calculate the differences in \( y \) as \( x \) increases by 1 unit:
- From \( x = 0 \) to \( x = 1 \): \( \Delta y = -2 - (-6) = 4 \)
- From \( x = 1 \) to \( x = 2 \): \( \Delta y = 0 - (-2) = 2 \)
- From \( x = 2 \) to \( x = 3 \): \( \Delta y = 1 - 0 = 1 \)
- From \( x = 3 \) to \( x = 4 \): \( \Delta y = 0 - 1 = -1 \)

### Step 2: Analyze the differences.
Since the change in \( y \) is not consistent (we get differences of 4, 2, 1, and -1), the function does **not** have a constant rate of change. Therefore, the function is **nonlinear**.

### Conclusion:
The function is **nonlinear** because the change in \( y \) values is not constant as \( x \) increases.

Would you like further details or have any questions?

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Here are 5 questions to expand on this topic:

1. What are the characteristics of a linear function?
2. How do you calculate the slope of a linear function?
3. Can a nonlinear function have a constant slope over a small interval?
4. What are some common forms of nonlinear functions?
5. How can you graphically determine if a function is linear or nonlinear?

**Tip:** Always check the consistency of the change in \( y \) values (the slope) to determine if a function is linear.

Determine whether the given function is linear or nonlinear. If it is linear, determine the slope.
x
y = f(x)
0 -6
1 -2
2 0
3 1
4 0

Explore how to determine if a function is linear or nonlinear by checking the consistency of the rate of change. This example involves calculating differences in y-values and uses the slope formula to identify nonlinear behavior in the function. Suitable for students in Grades 8-10.

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