Let's analyze the question step-by-step:
1. Understanding the Parabola:
- A parabola is a symmetric curve that is defined by a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- The shape of the parabola (whether it opens up or down) is determined by the coefficient [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
2. Vertex of the Parabola:
- The vertex is the highest or lowest point on the parabola.
- For a parabola that opens upwards (when [tex]\( a > 0 \)[/tex]), the vertex represents the minimum point.
- For a parabola that opens downwards (when [tex]\( a < 0 \)[/tex]), the vertex represents the maximum point.
3. Finding the y-value of the Vertex:
- The coordinates of the vertex for the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] are given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting this [tex]\( x \)[/tex]-value back into the equation gives the y-value of the vertex.
- However, because we're interested in the general property and not in calculating it specifically for a given equation, understanding that it is the minimum or maximum point is enough.
4. Choices Analysis:
- A. Maximum value: This would be true if the parabola opened downwards. Since we're discussing a parabola that opens upwards, this is not correct.
- B. Minimum value: This is correct because the vertex of a parabola that opens upwards is the lowest point, making it the minimum value.
- C. Axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex, defined by [tex]\( x = -\frac{b}{2a} \)[/tex]. This is not the y-value of the vertex.
- D. Quadratic function: This refers to the whole equation defining the parabola, not specifically the y-value of the vertex.
Therefore, the correct answer is:
B. minimum value.
