To find the range of the quadratic function $y = (x + 4)^2 - 5$, let's analyze its structure.

Step-by-Step Solution:

1. Identify the form of the function: This function is in the vertex form $y = a(x - h)^2 + k$, where $a = 1$, $h = -4$, and $k = -5$.

2. Determine the direction of the parabola: Since $a = 1 > 0$, the parabola opens upwards.

3. Find the vertex: The vertex of the function $y = (x + 4)^2 - 5$ is located at $(-4, -5)$. This point represents the minimum point of the parabola.

4. Analyze the range: Since the parabola opens upwards, the minimum value of $y$ is $-5$, and $y$ can take all values greater than or equal to $-5$. Therefore, the range of $y$ is $y \geq -5$.

Conclusion:

The correct answer is:

• D. $\{ y \, | \, y \geq -5 \}$

Would you like more details or have any questions?

Related Questions:

1. How do you determine the vertex of a quadratic function in vertex form?
2. What are the differences between vertex form and standard form of a quadratic function?
3. Why does the sign of $a$ (the leading coefficient) affect the direction of the parabola?
4. How do you find the axis of symmetry for a quadratic function in vertex form?
5. What is the range of a downward-opening parabola with a vertex of $(h, k)$?

Tip:

For any quadratic function in vertex form, the vertex point gives you the minimum or maximum value, which helps in determining the range quickly.