Certainly! Let's analyze the given quadratic equation [tex]\( y = -(x+6)^2 + 6 \)[/tex] step-by-step to determine the nature of its graph.
1. Equation Form:
The equation is given in the form [tex]\( y = -(x+6)^2 + 6 \)[/tex]. This is a standard quadratic equation of the form [tex]\( y = a(x-h)^2 + k \)[/tex], where:
- [tex]\(a\)[/tex] is the coefficient of the squared term,
- [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are the coordinates of the vertex of the parabola.
2. Identify the Vertex:
In the given equation, [tex]\( y = -(x+6)^2 + 6 \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( h = -6 \)[/tex]
- [tex]\( k = 6 \)[/tex]
Therefore, the vertex of the parabola is at the point [tex]\( (h, k) = (-6, 6) \)[/tex].
3. Direction of the Parabola:
Since the coefficient [tex]\( a = -1 \)[/tex] is negative, the parabola opens downwards. When the coefficient of the [tex]\( (x - h)^2 \)[/tex] term is negative, the graph of the parabola is an upside-down U-shape.
4. Nature of the Vertex:
For a downward-opening parabola, the vertex represents the maximum point of the graph.
5. Conclusion:
Since the vertex is at [tex]\( (-6, 6) \)[/tex] and it is a maximum point, the correct description of the graph is:
[tex]\( \boxed{\text{D. Maximum at } (-6, 6)} \)[/tex]
