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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?

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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.

Let the graph of g be a horizontal shrink by a factor of 2/3, followed by a translation 5 units left and 2 units down of the graph of f(x) = x^2. Write a rule for g.

Explore how a quadratic function f(x) = x^2 undergoes transformations to become g(x) through a horizontal shrink by a factor of 2/3 and translations 5 units left and 2 units down. This detailed guide provides insights into function transformations applicable in high school curriculum.

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