Let's analyze the quadratic function step-by-step to determine the correct statement regarding its vertex.
Given the function:
[tex]\[ m(x) = -(x+1)^2 + 4 \][/tex]
### Step 1: Identify the Vertex
A quadratic function of the form [tex]\( m(x) = a(x-h)^2 + k \)[/tex] has its vertex at the point [tex]\((h, k)\)[/tex].
Comparing [tex]\( m(x) = -(x+1)^2 + 4 \)[/tex] with the standard form [tex]\( a(x-h)^2 + k \)[/tex]:
- The expression [tex]\((x+1)\)[/tex] can be written as [tex]\((x - (-1))\)[/tex]. Hence, [tex]\( h = -1 \)[/tex].
- The constant term [tex]\( +4 \)[/tex] represents [tex]\( k \)[/tex].
So, the vertex of the quadratic function [tex]\( m(x) = -(x+1)^2 + 4 \)[/tex] is at the point [tex]\((-1, 4)\)[/tex].
### Step 2: Determine Whether the Vertex is a Maximum or Minimum
The sign of the coefficient [tex]\( a \)[/tex] in the quadratic function [tex]\( a(x-h)^2 + k \)[/tex] determines if the vertex is a maximum or minimum.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, and the vertex is a minimum point.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, and the vertex is a maximum point.
In the given function [tex]\( m(x) = -(x+1)^2 + 4 \)[/tex], the coefficient of [tex]\((x+1)^2\)[/tex] is [tex]\(-1\)[/tex], which is less than zero ([tex]\(-1 < 0\)[/tex]). Therefore, the parabola opens downwards, and the vertex represents a maximum point.
### Conclusion
Considering the vertex [tex]\((-1, 4)\)[/tex] and that the parabola opens downwards, the correct statement is:
- Its vertex is a maximum and is located at [tex]\((-1, 4)\)[/tex].
Thus, the correct choice is:
Its vertex is a maximum and is located at [tex]\((-1, 4)\)[/tex].
