To understand the properties of a parabola that has the same point as its y-intercept and the reflection point to the y-intercept, let's break it down step-by-step:
1. Definition of a Y-Intercept: The y-intercept of a parabola is the point where the parabola crosses the y-axis. This is generally denoted as the point (0, c).
2. Reflection Point and Symmetry: In the context of a parabola, symmetry plays a crucial role. A parabola has a vertical line called the axis of symmetry, which divides the parabola into two mirror-image halves. The vertex, also known as the turning point or reflection point, is situated on this axis of symmetry.
3. Vertex on the Y-Axis: If a parabola has its reflection point (vertex) as the same point as its y-intercept, it implies that the vertex lies on the y-axis.
4. Parabola's Equation: The general form of a parabolic equation is [tex]\( y = ax^2 + bx + c \)[/tex].
- Here, the term [tex]\( bx \)[/tex] vanishes when the vertex lies on the y-axis, simplifying to [tex]\( y = ax^2 + c \)[/tex].
- This equation indicates that the parabola is symmetric about the y-axis.
In summary, for a parabola that has its y-intercept as the reflection point (vertex), the vertex lies on the y-axis. Therefore, its equation simplifies to the form [tex]\( y = ax^2 + c \)[/tex], implying symmetry about the y-axis. This characteristic ensures the parabola opens either upwards or downwards depending on the sign of the coefficient [tex]\( a \)[/tex].
