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The given problem asks for the transformation of the function \( f(x) = x^2 \) into a new function \( g(x) \), involving three transformations:

1. **Vertical shrink by a factor of \( \frac{1}{3} \)**.
2. **Reflection in the y-axis**.
3. **Translation 3 units to the right**.

Let's break down the transformation step by step:

### Step 1: Vertical Shrink by \( \frac{1}{3} \)
A vertical shrink multiplies the entire function by \( \frac{1}{3} \), so the function becomes:
\[
g(x) = \frac{1}{3} f(x) = \frac{1}{3} x^2
\]

### Step 2: Reflection in the y-axis
A reflection in the y-axis replaces \( x \) with \( -x \). Applying this reflection:
\[
g(x) = \frac{1}{3} (-x)^2 = \frac{1}{3} x^2
\]
Note that reflecting \( x^2 \) about the y-axis doesn’t change the function because \( (-x)^2 = x^2 \).

### Step 3: Translation 3 units to the right
To translate the function 3 units to the right, replace \( x \) with \( x - 3 \). The equation now becomes:
\[
g(x) = \frac{1}{3} (x - 3)^2
\]

### Vertex
The vertex of \( f(x) = x^2 \) is at \( (0, 0) \). After translating 3 units to the right, the new vertex is at \( (3, 0) \).

### Final Answer
The rule for \( g(x) \) is:
\[
g(x) = \frac{1}{3} (x - 3)^2
\]
The vertex is \( (3, 0) \).

Would you like further details on any of the steps?

Here are 5 related questions to extend your understanding:
1. How does a vertical shrink differ from a vertical stretch?
2. What is the effect of translating a function vertically instead of horizontally?
3. How do you find the vertex of any quadratic function in the form \( a(x - h)^2 + k \)?
4. What happens to the graph of \( f(x) = x^2 \) when reflected over the x-axis instead of the y-axis?
5. Can a function be reflected over both axes simultaneously?

**Tip:** When reflecting over the y-axis, quadratic functions often remain unchanged because \( x^2 = (-x)^2 \).

The graph of g is a vertical shrink by a factor of 1/3 and a reflection in the y-axis, followed by a translation 3 units right of the graph of f(x) = x^2. Write a rule for g. Then identify the vertex.

This problem explores quadratic transformations, including vertical shrink by a factor of 1/3, reflection over the y-axis, and horizontal translation. Learn how to transform the function f(x) = x^2 into g(x) and determine the new vertex.

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