Alright, let's break down the problem step by step to determine the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for the given parabolic equation [tex]\( y = a(x-b)^2 + c \)[/tex]:
1. Understanding 'a':
- The coefficient [tex]\(a\)[/tex] determines the direction the parabola opens. If [tex]\(a\)[/tex] is positive, the parabola opens upwards, indicating a minimum point. If [tex]\(a\)[/tex] is negative, the parabola opens downwards, indicating a maximum point.
- Since the question states that the parabola has a minimum plane, [tex]\(a\)[/tex] must be positive.
- Therefore, the possible value for [tex]\(a\)[/tex] is [tex]\(a = 1\)[/tex].
2. Understanding 'b':
- The parameter [tex]\(b\)[/tex] indicates the horizontal shift of the parabola. It does not affect whether the parabola has a minimum or maximum, only its position along the x-axis.
- The question provides two possible values for [tex]\(b\)[/tex]: [tex]\(b = 2\)[/tex] and [tex]\(b = -2\)[/tex].
- Both values are valid and do not violate any conditions of the problem.
3. Understanding 'c':
- The coefficient [tex]\(c\)[/tex] indicates the vertical shift of the parabola. Like [tex]\(b\)[/tex], it does not affect the direction in which the parabola opens, it only shifts the parabola up or down on the y-axis.
- The problem does not provide specific values for [tex]\(c\)[/tex], but it can be any real number since it simply moves the parabola vertically without changing its shape or the direction it opens.
Summing up, the correct values are:
- [tex]\(a = 1\)[/tex] (because the parabola opens upwards to have a minimum)
- [tex]\(b = 2\)[/tex] and [tex]\(b = -2\)[/tex] (either value is acceptable as it only represents a horizontal shift)
For [tex]\(c\)[/tex], it can take any real value, but since this question only allows selection from the given choices of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], it is not directly mentioned but implied.
Therefore, the selections you should make are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(b = -2\)[/tex]
