To determine whether the given equations $y = 3x$ and $y = -6x$ represent linear or nonlinear functions, we analyze their form and properties:

1. Equation $y = 3x$:

   • This equation is in the form $y = mx + b$, where $m$ (the slope) is $3$ and $b$ (the y-intercept) is $0$.
   • Equations in this form represent straight lines, meaning the relationship between $x$ and $y$ is linear.
   • Therefore, $y = 3x$ is a linear function.

2. Equation $y = -6x$:

   • Similarly, this equation is also in the form $y = mx + b$, where $m$ is $-6$ and $b$ is $0$.
   • The graph of this equation is a straight line with a negative slope.
   • Thus, $y = -6x$ is also a linear function.

Explanation

Both equations follow the general linear equation format $y = mx + b$, which represents straight-line relationships. This makes them linear functions. The defining characteristic of a linear function is that its graph forms a straight line, and it has a constant rate of change (slope).

Would you like additional details or examples? Here are related questions to explore further:

1. What distinguishes linear functions from nonlinear functions graphically?
2. How does the slope $m$ affect the steepness and direction of a linear graph?
3. Can the y-intercept $b$ in $y = mx + b$ ever make the function nonlinear?
4. What are some examples of nonlinear functions, and how are they identified algebraically?
5. How do real-life situations model linear versus nonlinear relationships?

Tip: Always check for the power of $x$. If $x$ has an exponent other than 1 or if $x$ appears in a denominator, the function is likely nonlinear.