Let's go through each function to determine if it is linear or nonlinear:

1. $y = 2x^3$

• This is a nonlinear function because it involves $x^3$, which is a cubic term. Linear functions only have terms where the variable is raised to the first power.

2. $y = x + 5$

• This is a linear function. It is in the form $y = mx + b$, where $m = 1$ (the slope) and $b = 5$ (the y-intercept). Linear functions have a straight-line graph.

3. $y = x^2 - 3$

• This is a nonlinear function because it includes $x^2$, which is a quadratic term.

4. $y = \frac{x \cdot 2}{2}$

• This simplifies to $y = x$, which is a linear function because it is a straight line with a slope of 1.

5. $5x + 5y = 25$

• This is a linear function. It can be rewritten in the slope-intercept form as $y = -x + 5$, which is a straight line.

Summary:

• Linear functions:
  $y = x + 5$, $y = \frac{x \cdot 2}{2}$, $5x + 5y = 25$

• Nonlinear functions:
  $y = 2x^3$, $y = x^2 - 3$

Would you like a further breakdown or clarification on any of these?
Here are some additional questions to explore:

1. What defines a linear function mathematically?
2. How can you graph linear vs. nonlinear functions?
3. What are some real-world examples of linear and nonlinear relationships?
4. How do nonlinear functions differ in their graphs from linear functions?
5. What is the importance of understanding linearity in mathematical modeling?

Tip: Linear equations form straight lines when graphed, whereas nonlinear equations can form curves, parabolas, or other shapes.