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To solve this problem, we need to apply two transformations to the function \( f(x) = \frac{2}{3}(x - 4)^2 - 6 \):

1. **Translate left by 3 units**: This means replacing \( x \) with \( x + 3 \).
2. **Translate up by 2 units**: This means adding 2 to the entire function.

### Step-by-Step Solution

Given:
\[
f(x) = \frac{2}{3}(x - 4)^2 - 6
\]

1. **Translate left by 3 units**:
   \[
   f(x + 3) = \frac{2}{3}((x + 3) - 4)^2 - 6
   \]
   Simplify inside the parentheses:
   \[
   f(x + 3) = \frac{2}{3}(x - 1)^2 - 6
   \]

2. **Translate up by 2 units**:
   \[
   g(x) = f(x + 3) + 2 = \frac{2}{3}(x - 1)^2 - 6 + 2
   \]
   Simplify the constant terms:
   \[
   g(x) = \frac{2}{3}(x - 1)^2 - 4
   \]

### Final Answer
The quadratic equation \( g(x) \) representing \( f(x) \) translated left 3 units and up 2 units is:
\[
g(x) = \frac{2}{3}(x - 1)^2 - 4
\]

Would you like further details on this solution?

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Here are five related questions to expand your understanding:

1. How would you translate a function \( f(x) \) right by 5 units and down by 3 units?
2. How does the vertex of \( f(x) = a(x - h)^2 + k \) change with translations?
3. What effect does changing the coefficient \( a \) have on the shape of a quadratic function?
4. How do you reflect a quadratic function \( f(x) \) across the x-axis?
5. How would you express a quadratic function in standard form after applying transformations?

**Tip:** When translating functions horizontally, remember that left and right shifts change the \( x \)-variable inside the parentheses.

The quadratic function f(x) = (2/3)(x - 4)^2 - 6 is given. Write the quadratic equation g(x) that represents f(x) translated left 3 units and up two units.

Learn how to translate the quadratic function f(x) = (2/3)(x - 4)^2 - 6 by 3 units to the left and 2 units up. This solution provides a step-by-step transformation, covering algebraic concepts suitable for Grades 9-10.

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