Let's break down the transformation of the linear parent function $f(x) = x$ to the function $g(x) = \frac{1}{3}x - 2$.

1. Slope Comparison: The slope of the parent function $f(x) = x$ is 1. In the transformed function $g(x) = \frac{1}{3}x - 2$, the slope is $\frac{1}{3}$, which is less steep than 1. So, the graph of $g(x)$ is less steep than $f(x)$.

2. Y-Intercept Shift: The parent function $f(x) = x$ has a y-intercept of 0. The function $g(x) = \frac{1}{3}x - 2$ has a y-intercept of -2. This means the graph of $g(x)$ is shifted downward by 2 units compared to $f(x)$.

Given these two observations, the correct comparison is:

• The graph of $g(x)$ is less steep than the graph of $f(x)$, and the y-intercept has been shifted down.

Therefore, the correct answer is A.

Would you like further clarification on this? Feel free to ask!

Here are some related questions for further exploration:

1. How does changing the slope affect the steepness of a linear function?
2. What happens to the graph of a linear function when the y-intercept changes?
3. How do horizontal shifts affect the graph of a linear function?
4. How would the graph change if the function had a slope greater than 1?
5. What does a negative slope do to the graph of a linear function?

Tip: Always check the slope and the y-intercept of the transformed function to understand the graph's behavior!