To find the rule for the function $g(x)$, which is a transformation of $f(x) = x^2$, we need to apply the given transformations in the correct order.

Step 1: Horizontal Shrink by a Factor of 1/3

A horizontal shrink by a factor of $1/3$ is applied by replacing $x$ with $3x$. So, we first transform the function $f(x) = x^2$ to: $f(3x) = (3x)^2 = 9x^2$ This gives the horizontally shrunk version of the function.

Step 2: Translation 1 Unit Up

To translate the graph 1 unit upward, we add 1 to the function: $g(x) = 9x^2 + 1$

Thus, the rule for $g(x)$ is: $g(x) = 9x^2 + 1$

Would you like further explanation or have any questions?

Here are 5 related questions:

1. What effect does a horizontal shrink have on a graph?
2. How would the rule change if the translation was 2 units up instead?
3. What if instead of a shrink, there was a horizontal stretch by a factor of 3?
4. How does translating a graph vertically differ from horizontal translation?
5. How would the rule for $g(x)$ change with a reflection over the x-axis?

Tip: When applying multiple transformations to a function, follow the sequence: horizontal changes (shrink/stretch), reflections, and then translations.