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!