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?

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.