1. **State the problem:** We are given the quadratic function $$y = -(x + 5)^2 + 4$$ and need to identify its maximum or minimum value, determine whether it is a maximum or minimum, find the axis of symmetry, and find the domain and range. 2. **Formula and important rules:** A quadratic function in vertex form is $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex. - If $$a < 0$$, the parabola opens downward and the vertex is a maximum point. - If $$a > 0$$, the parabola opens upward and the vertex is a minimum point. - The axis of symmetry is the vertical line $$x = h$$. - The domain of any quadratic function is all real numbers. - The range depends on the vertex and whether the parabola opens up or down. 3. **Identify vertex:** The given function is $$y = -(x + 5)^2 + 4$$ which can be rewritten as $$y = -(x - (-5))^2 + 4$$. - So, the vertex is at $$(-5, 4)$$. 4. **Determine maximum or minimum:** Since $$a = -1 < 0$$, the parabola opens downward. - Therefore, the vertex represents a **maximum** value. 5. **Axis of symmetry:** The axis of symmetry is the vertical line through the vertex's x-coordinate: $$x = -5$$. 6. **Domain:** The domain of any quadratic function is all real numbers: $$\text{Domain} = (-\infty, \infty)$$. 7. **Range:** Since the parabola opens downward and the maximum value is 4 at $$x = -5$$, the range is all $$y$$ values less than or equal to 4: $$\text{Range} = (-\infty, 4]$$. **Final answers:** - Maximum value: $$4$$ - It is a maximum - Axis of symmetry: $$x = -5$$ - Domain: all real numbers - Range: $$y \leq 4$$