The problem involves transforming the graph of $f(x) = x^2$ through two steps: a horizontal shrink and a translation.

Step 1: Horizontal Shrink

The horizontal shrink is by a factor of $\frac{2}{3}$. A horizontal shrink by this factor means you replace $x$ with $\frac{3}{2}x$ in the function, because a horizontal shrink by $k$ changes $f(x)$ to $f\left( \frac{x}{k} \right)$.

So, the new function after the horizontal shrink is: $f\left( \frac{3}{2}x \right) = \left( \frac{3}{2}x \right)^2 = \frac{9}{4}x^2.$

Step 2: Translation

• The translation is 5 units left and 2 units down.
• To translate a function horizontally to the left by 5 units, we replace $x$ with $x + 5$.
• To translate a function vertically down by 2 units, we subtract 2 from the whole function.

Thus, applying the translation to the horizontally shrunk function gives: $g(x) = \frac{9}{4}(x + 5)^2 - 2.$

Final Rule for $g(x)$:

$g(x) = \frac{9}{4}(x + 5)^2 - 2.$

This is the rule for the transformed function $g(x)$.

Do you want further details or have any questions?

Here are 5 related questions to extend this:

1. What happens if we apply a vertical stretch instead of a horizontal shrink?
2. How would the function change if the translation was 5 units right instead of left?
3. How does the factor $\frac{9}{4}$ affect the shape of the parabola compared to the original $f(x) = x^2$?
4. What is the effect of vertical shifts on the vertex of a parabola?
5. Can you describe the general formula for any transformations on $f(x) = x^2$?

Tip: Horizontal transformations involve changing the argument inside the function, while vertical transformations affect the entire function itself.