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 $(h, k)$.