Let's break this down step by step:

1. Original Function:

The original function is $f(x) = 2(x - 5)^2 + 3$.

2. Reflection Across the y-axis:

To reflect a function across the y-axis, you replace $x$ with $-x$ in the function. So:

$f(-x) = 2(-x - 5)^2 + 3$

Simplify inside the parentheses:

$f(-x) = 2(-x + 5)^2 + 3$

3. Translation Right by 6:

To translate a function right by 6 units, replace $x$ with $x - 6$. So:

$f(-x - 6) = 2((-x - 6) + 5)^2 + 3$

Simplify inside the parentheses:

$f(-x - 6) = 2((-x - 1)^2) + 3$

4. Translation Down by 2:

To translate a function down by 2 units, subtract 2 from the entire function. So:

$f(-x - 6) - 2 = 2((-x - 1)^2) + 3 - 2$

Simplify:

$f(-x - 6) - 2 = 2((-x - 1)^2) + 1$

Final Transformed Function:

The final equation after reflecting across the y-axis, translating right by 6 units, and down by 2 units is:

$f(x) = 2((-x - 1)^2) + 1$

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Would you like a detailed explanation of each step, or do you have any specific questions?

Related Questions:

1. How would the function change if it were reflected across the x-axis instead?
2. What would happen if the function were translated left instead of right?
3. Can you describe the vertex shift of the original function after these transformations?
4. How would you determine the axis of symmetry for the transformed function?
5. What effect does stretching or compressing have on these types of transformations?

Tip: Reflections across the y-axis flip the horizontal orientation, while translations shift the graph without changing its shape.