The given function is $f(x) = x^2$. To transform it based on the instructions:

1. Stretch vertically by a factor of 6: This multiplies the entire function by 6. The new function becomes $6x^2$.

2. Shift downward by 5 units: Subtract 5 from the function, so it becomes $6x^2 - 5$.

3. Shift 2 units to the right: Replace $x$ with $(x - 2)$ to shift the function horizontally. The final equation is:

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

This is the transformed function after applying the specified transformations.

Would you like further details or explanations?

5 Related Questions:

1. How would a horizontal stretch affect the function?
2. What happens if the function is shifted left instead of right?
3. How does a reflection across the x-axis alter the function?
4. Can you apply multiple transformations in a different order and get the same result?
5. What would happen if the function were shifted upward instead of downward?

Tip:

When shifting horizontally, replacing $x$ with $(x - h)$ moves the graph to the right, and replacing it with $(x + h)$ moves it to the left.