https://file.aiquickdraw.com/mathbot/file/17270009545262vlxq1y3.jpg

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.

Gambarkan grafik persamaan berikut:

a. y = x^2 + 5
b. y = √(x + 1) - 2

Learn how to graph the quadratic function y = x^2 + 5 and the square root function y = √(x + 1) - 2. This tutorial covers key graph transformations, such as shifting and translating graphs. Suitable for students in Grades 9-10.

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