To modify the graph of $y = (x + 3)^2 + 6$, different transformations (vertical and horizontal) can be applied as follows:

1. To translate the graph upwards by 2 units: $y = (x + 3)^2 + 6 + 2 = (x + 3)^2 + 8$

2. To translate the graph leftwards by 2 units: $y = \left( (x + 3) + 2 \right)^2 + 6 = (x + 5)^2 + 6$

3. To translate the graph rightwards by 2 units: $y = \left( (x + 3) - 2 \right)^2 + 6 = (x + 1)^2 + 6$

4. To translate the graph downwards by 2 units: $y = (x + 3)^2 + 6 - 2 = (x + 3)^2 + 4$

These transformations directly modify the constants associated with the graph's position.

Would you like further details on these transformations or have any questions?

Here are 5 related questions:

1. How would you translate the graph by 2 units diagonally (both left and down)?
2. How does translating a graph horizontally differ from translating it vertically?
3. What effect would multiplying the quadratic by a constant have on the graph?
4. How would a reflection over the x-axis affect the equation?
5. How would the equation change if you wanted to stretch the graph vertically by a factor of 2?

Tip: When translating horizontally, modifying $x$ inside the parentheses affects the left-right movement, while adjusting constants outside affects vertical shifts.