Identifying Linear Functions: Expressions for Parallel Lines

To solve this problem, we'll follow these steps:

• Step 1: Write each given expression in the slope-intercept form $y = mx + b$.
• Step 2: Identify the slope ($m$) of each expression.
• Step 3: Determine which expressions have the same slope, thus indicating parallel lines.

Now, let's analyze each choice:

Choice A:

The expressions are $y = 3x + 2$ and $y = 3(x + 2)$. Start by expanding the second equation:

$y = 3(x + 2) = 3x + 6$.

Both expressions $y = 3x + 2$ and $y = 3x + 6$ are linear functions, and they both have a slope $m = 3$. Thus, these lines are parallel.

Choice B:

The expressions are $y = 2x + 1$ and $y = x + 1$. Here, the first equation is a line with $m = 2$ and the second is a line with $m = 1$. These lines are not parallel as their slopes differ.

Choice C:

The expressions are $y = 2(x^2 + 1)$ and $y = 2x^2 + 2$. These expressions are quadratic, not linear, and therefore cannot be considered for parallel linear functions.

Choice D:

The expressions are $y = 2(x + 1)$ and $y = 3 + 2x$. Simplifying the first gives us:

$y = 2x + 2$.

Thus, both equations are $y = 2x + 2$ and $y = 2x + 3$ which are linear with slope $m = 2$. These lines are parallel.

Conclusion: The correct answer is "Answers A+D are correct" as both choices A and D consist of linear functions with parallel lines.