The given question asks to graph the function $f(x) = (x + 5)^2 - 4$ by using transformations starting from the parent function $y = x^2$. Let's walk through the steps of how this transformation works:

Transformations Breakdown:

1. Start with the parent function:
   $y = x^2$.
   This is a standard parabola that opens upwards, with the vertex at $(0, 0)$.

2. Horizontal Shift:
   The function $f(x) = (x + 5)^2 - 4$ has a horizontal shift.

   • $x + 5$ shifts the graph 5 units to the left (because the shift is opposite to the sign inside the parentheses).

3. Vertical Shift:
   The term $-4$ moves the graph 4 units down.

   • So, the new vertex will be at $(-5, -4)$.

4. No Reflection or Stretch/Compression:

   • The coefficient inside the squared term is 1, so the shape (width) of the parabola remains the same as the parent function.
   • There is no reflection (as there is no negative sign).

Summary of Transformations:

• Shift left by 5 units: $x + 5$.
• Shift down by 4 units: $-4$.
• No stretching/compression or reflection.

Thus, the vertex of the transformed parabola will be at (-5, -4), and it will maintain the same "U" shape as the original $y = x^2$.

Graphing Instructions:

1. Plot the vertex at $(-5, -4)$.
2. Since the graph is not stretched or compressed, follow the usual pattern:
   • From the vertex, move 1 unit right/left and 1 unit up to get additional points.
   • Then move 2 units right/left and 4 units up (since $2^2 = 4$).
3. Draw a smooth curve passing through these points.

Would you like me to create an image of this graph, or do you need more clarifications?

Related Questions:

1. What happens if we change $(x + 5)^2 - 4$ to $-(x + 5)^2 - 4$?
2. How do horizontal and vertical shifts affect the domain and range of a function?
3. What would happen if we introduced a stretching factor, like $2(x + 5)^2 - 4$?
4. How would the graph change if the equation was $(x + 5)^2 + 4$ instead?
5. What is the algebraic process to find the vertex of any quadratic function in vertex form?

Tip: When dealing with transformations, always check the sign inside the parentheses for horizontal shifts—it behaves opposite to intuition.