To compare the graphs of $f(x) = 2x + 1$ and $g(x) = 2(x + 3) + 1$, let's analyze the transformation involved.

1. Start with $f(x) = 2x + 1$: This is a linear function with a slope of 2 and a y-intercept of 1.

2. Analyze $g(x) = 2(x + 3) + 1$: Distribute the $2$ inside the parentheses: $g(x) = 2x + 6 + 1 = 2x + 7$

3. Identify the transformation: The resulting equation, $g(x) = 2x + 7$, shows that $g(x)$ is a vertical shift of $f(x)$. Specifically, $g(x)$ is $f(x)$ shifted up by 6 units because the constant term changed from $+1$ to $+7$.

Answer: The correct option is B. The graph of $g$ is the graph of $f$ translated 3 units up.

Would you like further details on graph transformations or have any questions?

Here are 5 related questions that expand on this concept:

1. How do horizontal translations affect the function’s equation?
2. How does a vertical stretch or compression alter the graph of a linear function?
3. What effect does reflecting a graph over the x-axis have on the function’s equation?
4. How can we determine a horizontal shift in more complex functions, like quadratics?
5. How do translations differ between linear and non-linear functions?

Tip: For linear functions, translations only alter the y-intercept when moving vertically, and affect the x-term inside parentheses when moving horizontally.