Consider the functions:

f(x) = x^2

g(x) = (x + 1)^2 – 2

h(x) = (x + 3)^2 + 4

Let's analyze the functions you've provided:

1. **Function \( f(x) = x^2 \)**:
   - This is a basic quadratic function, representing a parabola that opens upwards with its vertex at the origin (0, 0).

2. **Function \( g(x) = (x + 1)^2 - 2 \)**:
   - This can be rewritten as \( g(x) = x^2 + 2x + 1 - 2 = x^2 + 2x - 1 \).
   - It is also a parabola opening upwards, with a vertex shifted to the left by 1 unit and down by 2 units, located at \( (-1, -2) \).

3. **Function \( h(x) = (x + 3)^2 + 4 \)**:
   - This can be expanded to \( h(x) = x^2 + 6x + 9 + 4 = x^2 + 6x + 13 \).
   - It is a parabola that opens upwards, with its vertex shifted to the left by 3 units and up by 4 units, located at \( (-3, 4) \).

### Summary of Characteristics:
- All three functions are parabolas that open upwards.
- Each function has a vertex:
  - \( f(x) \) at \( (0, 0) \)
  - \( g(x) \) at \( (-1, -2) \)
  - \( h(x) \) at \( (-3, 4) \)

### Questions or Further Analysis:
1. Would you like to find the intersections of these functions?
2. Do you want to analyze their behavior as \( x \) approaches infinity?
3. Would you like to sketch the graphs of these functions?
4. Do you want to calculate specific function values at certain \( x \) points?
5. Would you like to explore transformations between these functions?

### Tip:
To understand the effects of transformations on quadratic functions, remember that \( (x - h)^2 + k \) shifts the vertex to \( (h, k) \).

Consider the functions: f(x) = x^2, g(x) = (x + 1)^2 – 2, h(x) = (x + 3)^2 + 4

Explore the characteristics of the quadratic functions f(x) = x^2, g(x) = (x + 1)^2 – 2, and h(x) = (x + 3)^2 + 4. This analysis includes vertex locations, transformations, and graphical behavior. Suitable for students in Grades 7-9.

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