To analyze the quadratic function [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex], let's break down the problem into its components: identifying the vertex, the domain, and the range.
1. Vertex:
The function [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex] is in the standard form of a quadratic function [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where:
- [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are the coordinates of the vertex [tex]\((h, k)\)[/tex].
Comparing [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex] to the standard form [tex]\( f(x) = a(x - h)^2 + k \)[/tex]:
- [tex]\( h = 2 \)[/tex]
- [tex]\( k = 4 \)[/tex]
Therefore, the vertex of the function [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex] is [tex]\((2, 4)\)[/tex].
2. Domain:
Quadratic functions are polynomial functions, and all polynomial functions have a domain of all real numbers. Hence, the domain of [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex] is all real numbers.
3. Range:
- Since the coefficient of the squared term [tex]\((x - 2)^2\)[/tex] is positive (coefficient [tex]\(a = 1\)[/tex]), the parabola opens upwards.
- For a parabola that opens upwards, the range is all [tex]\( y \)[/tex]-values greater than or equal to the [tex]\( y \)[/tex]-value of the vertex.
Given that the [tex]\( y \)[/tex]-coordinate of the vertex is [tex]\( 4 \)[/tex], the range of the function [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex] is [tex]\( y \geq 4 \)[/tex].
So, the correct identification for the vertex, domain, and range of the function [tex]\( f(x) = (x - 2)^2 + 4 \)[/tex] is:
- The vertex is [tex]\((2, 4)\)[/tex].
- The domain is all real numbers.
- The range is [tex]\( y \geq 4 \)[/tex].
Thus, the correct answer is:
The vertex is [tex]\((2, 4)\)[/tex], the domain is all real numbers, and the range is [tex]\( y \geq 4 \)[/tex].
