In mathematics, a function is periodic if it repeats its values at regular intervals. More formally, a function \( f(x) \) is periodic with period \( T \) if:

\[
f(x + T) = f(x)
\]

for all \( x \), where \( T > 0 \).

Now, let's check if the given function \( y = -2x \) is periodic.

1. The function is \( y = -2x \).
2. Assume this function is periodic with some period \( T \). Then, we need to verify if:

\[
-2(x + T) = -2x
\]

3. Simplifying the left-hand side:

\[
-2(x + T) = -2x - 2T
\]

4. For the function to be periodic, the equation must hold true:

\[
-2x - 2T = -2x
\]

This simplifies to:

\[
-2T = 0
\]

Which implies that \( T = 0 \). A period of 0 doesn't make sense, since \( T > 0 \) by definition.

### Conclusion:
The function \( y = -2x \) is **not periodic**, because it does not repeat its values at any non-zero interval. Instead, it is a linear function, which keeps decreasing as \( x \) increases and never repeats.

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Do you want more details or have any questions?

Here are 5 related questions:
1. What are examples of periodic functions?
2. How can we determine the period of a trigonometric function like \( \sin(x) \)?
3. What are the characteristics of linear functions like \( y = -2x \)?
4. Can non-linear functions be periodic?
5. How does the slope of a linear function affect its graph?

**Tip:** Linear functions like \( y = mx + b \) are generally not periodic because they increase or decrease indefinitely without repeating values.

Explore why the linear function y = -2x is not periodic. Understand the properties of linear functions and how they differ from periodic functions. This detailed explanation covers algebraic concepts and is suitable for students in Grades 9-12.

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