To identify the vertex of the parabola described by the equation [tex]\( y = (x-5)^2 - 2 \)[/tex], we can utilize the vertex form of a parabolic equation, which is [tex]\( y = a(x-h)^2 + k \)[/tex]. In this form, the vertex of the parabola corresponds to the point [tex]\((h, k)\)[/tex].
Let's break down the given equation [tex]\( y = (x-5)^2 - 2 \)[/tex]:
1. Identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
- The equation [tex]\( y = (x-5)^2 - 2 \)[/tex] is already in vertex form, where:
- [tex]\( h = 5 \)[/tex] (this is the value of [tex]\( x \)[/tex] that makes the term [tex]\((x-5)\)[/tex] equal to zero)
- [tex]\( k = -2 \)[/tex] (this is the constant term that adjusts the vertical position of the parabola with respect to the [tex]\( x \)[/tex]-axis)
2. Determine the vertex coordinates:
- Given the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex], the vertex of the parabola is at the point [tex]\( (h, k) \)[/tex].
- Therefore, substituting the identified values, we have [tex]\( (h, k) = (5, -2) \)[/tex].
3. Compare with the given choices:
- (5, -2)
- (5, 2)
- (-5, -2)
From our detailed analysis and the identification process, it is clear that the vertex of the parabola described by the equation [tex]\( y = (x-5)^2 - 2 \)[/tex] is [tex]\( (5, -2) \)[/tex].
Hence, the correct answer is:
[tex]\[ (5, -2) \][/tex]
