The problem asks to sketch the graphs of the following equations:

a. $y = x^2 + 5$

b. $y = \sqrt{x + 1} - 2$

Let's analyze and sketch each graph.

(a) $y = x^2 + 5$

This is a parabola that opens upwards with its vertex at $(0, 5)$. The graph of $y = x^2$ is shifted vertically up by 5 units. It will look like the standard parabola but starting from 5 units above the x-axis.

(b) $y = \sqrt{x + 1} - 2$

This is a square root function. The graph of $y = \sqrt{x}$ is shifted left by 1 unit and then down by 2 units. It starts at the point $(-1, -2)$, since the square root function is only defined for non-negative values of the argument inside it.

Steps to sketch the graphs:

1. For $y = x^2 + 5$:

   • Plot the vertex at $(0, 5)$.
   • Draw a parabola opening upwards.
   • Identify a few points like $(-1, 6)$, $(1, 6)$, and $(-2, 9)$, $(2, 9)$.

2. For $y = \sqrt{x + 1} - 2$:

   • Plot the starting point at $(-1, -2)$.
   • For $x = 0$, $y = \sqrt{0 + 1} - 2 = -1$, so the point is $(0, -1)$.
   • For $x = 3$, $y = \sqrt{3 + 1} - 2 = 0$, so the point is $(3, 0)$.
   • Draw the curve starting from $(-1, -2)$ and gradually increasing.

Would you like to see the graph sketches or have any further explanations?

Follow-up Questions:

1. What are the key transformations that apply to the graph of $y = x^2$ when changing it to $y = x^2 + 5$?
2. How does shifting a square root function horizontally and vertically affect its graph?
3. What is the domain and range of $y = \sqrt{x + 1} - 2$?
4. How do the vertex and symmetry properties change when constants are added to $x^2$ or $\sqrt{x}$ functions?
5. What are the intercepts of the given functions on the graph?

Tip: Always identify the basic shape of the function before applying any shifts or transformations. This helps in visualizing the final graph more accurately.