The question asks for the transformations needed to graph the function $h(x) = (x + 2)^2 + 5$ using the basic quadratic function $f(x) = x^2$ as a starting point.

We can break this down into two transformations:

1. Horizontal Shift: The term $(x + 2)^2$ represents a shift 2 units to the left. This is because adding $2$ inside the parentheses moves the graph in the negative $x$-direction.

2. Vertical Shift: The $+5$ outside the squared term represents a shift 5 units up. This shifts the graph upward by 5 units on the $y$-axis.

So, the required transformations are:

• A horizontal shift 2 units to the left.
• A vertical shift 5 units up.

Would you like more details or have any questions on this?

Here are 5 related questions for you to consider:

1. What happens to the graph if the constant inside the parentheses is negative, like $h(x) = (x - 3)^2 + 5$?
2. How would the graph change if the $+5$ were replaced with a negative value, like $h(x) = (x + 2)^2 - 3$?
3. How do horizontal and vertical stretches or compressions affect the transformation of $f(x) = x^2$?
4. Can you describe how to graph $h(x) = -(x+2)^2 + 5$ based on the transformations?
5. How would you graph $h(x) = 2(x+2)^2 + 5$, where the quadratic term is multiplied by a coefficient?

Tip: When applying transformations, always follow the order: horizontal transformations, then vertical transformations.