https://file.aiquickdraw.com/mathbot/file/1726109751535cbr7xdso.jpg

The given quadratic function is:

\[
y = (x + 5)^2 - 4
\]

To sketch the graph, let's break it down:

### 1. **Vertex Form**:
The equation is in vertex form, \( y = a(x - h)^2 + k \), where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the direction and width of the parabola.

Here:
- \( h = -5 \)
- \( k = -4 \)
Thus, the vertex is at \( (-5, -4) \).

### 2. **Direction**:
Since the coefficient of \( (x + 5)^2 \) is positive (i.e., \( a = 1 \)), the parabola opens **upwards**.

### 3. **Shifts**:
- The graph is shifted 5 units to the **left** and 4 units **down** from the standard parabola \( y = x^2 \).

### 4. **Axis of Symmetry**:
The axis of symmetry is the vertical line passing through the vertex, i.e., \( x = -5 \).

### 5. **Y-Intercept**:
When \( x = 0 \):
\[
y = (0 + 5)^2 - 4 = 25 - 4 = 21
\]
So the y-intercept is \( (0, 21) \).

### Plotting Steps:
- Mark the vertex at \( (-5, -4) \).
- Draw the axis of symmetry at \( x = -5 \).
- Since the parabola opens upwards, plot additional points by choosing values of \( x \) around the vertex (e.g., \( x = -4 \), \( x = -6 \)) to find corresponding \( y \)-values.
- Connect the points smoothly to form the parabola.

Would you like further details or have any questions? Here are some related questions to deepen your understanding:

1. What are the general properties of a quadratic function in vertex form?
2. How can we find the roots of \( y = (x + 5)^2 - 4 \)?
3. How would the graph change if the coefficient of \( (x + 5)^2 \) was negative?
4. What does the axis of symmetry represent geometrically?
5. Can we transform this function into standard quadratic form?

**Tip:** The vertex form of a quadratic function is especially useful for quickly identifying the vertex and the direction of the parabola!

Sketch the graph of the given quadratic function y = (x + 5)^2 - 4.

Learn how to sketch the graph of the quadratic function y = (x + 5)^2 - 4. This problem explores key concepts in graphing parabolas, including the vertex, axis of symmetry, and shifts in the graph. Suitable for students in Grades 9-11.

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