In order to determine the range of the function [tex]\( f(x) = (x + 4)^2 + 7 \)[/tex], we need to analyze the behavior of the quadratic function.
1. Identify the form of the function:
The given function is of the form [tex]\( f(x) = (x + 4)^2 + 7 \)[/tex].
2. Recognize the vertex form:
The function is in the vertex form of a quadratic function, which is generally written as [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola. In this case, [tex]\( a = 1 \)[/tex], [tex]\( h = -4 \)[/tex], and [tex]\( k = 7 \)[/tex].
3. Determine the vertex:
The vertex of the function [tex]\( (x + 4)^2 + 7 \)[/tex] is [tex]\( (-4, 7) \)[/tex].
4. Analyze the coefficient [tex]\( a \)[/tex]:
Since the coefficient of the squared term [tex]\( (a = 1) \)[/tex] is positive, the parabola opens upwards. This means the vertex represents the minimum point of the function.
5. Determine the minimum value:
The minimum value of the function occurs at the vertex, where [tex]\( y = 7 \)[/tex].
6. Determine the range:
Since the parabola opens upwards and the minimum value is [tex]\( y = 7 \)[/tex], the range of the function is all [tex]\( y \)[/tex]-values that are greater than or equal to 7.
Hence, the correct answer is:
[tex]\[ \boxed{y \geq 7} \][/tex]
This corresponds to option 4.
