When analyzing the intervals of increase and decrease of a parabola, it is crucial to consider the following factors:
1. The position of the vertex: The vertex of a parabola is the highest or lowest point, depending on whether the parabola opens upwards or downwards. It serves as a crucial pivot point that determines where the parabola transitions from increasing to decreasing or vice versa. By knowing the position of the vertex, you can accurately identify the interval where the parabola changes its direction.
2. The symmetry of the parabola about the y-axis: Parabolas are symmetric figures, and this symmetry helps in understanding their behavior on either side of the vertex. This symmetry can simplify the analysis because if you know the behavior of the parabola on one side of the vertex, you can infer the behavior on the other side due to symmetry.
4. The sign of the coefficient 'a': The coefficient 'a' in the quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] determines the direction the parabola opens. If [tex]\( a \)[/tex] is positive, the parabola opens upwards, meaning the vertex represents the minimum point, and the intervals will be increasing on both sides from the vertex. If [tex]\( a \)[/tex] is negative, the parabola opens downwards, meaning the vertex represents the maximum point, and the intervals will be decreasing on both sides from the vertex.
Therefore, the factors that should be considered are:
- The position of the vertex.
- The symmetry of the parabola about the y-axis.
- The sign of the coefficient 'a'.
Thus, the correct choices are:
- The position of the vertex.
- The symmetry of the parabola about the y-axis.
- The sign of the coefficient 'a'.
