To solve this problem, let's analyze the given information step by step.
1. Understand the Vertices and Orientation of Parabolas:
- The first parabola has a vertex at [tex]\((0, 4)\)[/tex] and it opens downward. This means its maximum point is at [tex]\((0, 4)\)[/tex] and from there it curves downward.
- The second parabola has a vertex at [tex]\((0, -4)\)[/tex]. The direction in which this parabola opens will affect how and where it intersects with the first parabola.
2. Intersection Points:
- For the two parabolas to intersect at two points, the curves must cross each other. Given that the first parabola opens downward and has its vertex at [tex]\((0, 4)\)[/tex], any downward opening curve from [tex]\((0, -4)\)[/tex] would move further down and thus, would not intersect the first parabola again.
- Therefore, for the second parabola to intersect the first one, it must open upward. This setup ensures the parabolas will meet at two points - one point above and one point below the x-axis.
3. Symmetrical Intersection:
- The points of intersection will be symmetrically located with respect to the y-axis because both vertices lie on the y-axis ([tex]\(x=0\)[/tex]). This symmetry about the y-axis implies that the parabola's shapes will create points of intersection at equal distances to either side of the y-axis.
4. Drawing Conclusions about Intersection:
- The first parabola opens downward.
- The second parabola needs to open upward to ensure that they intersect at exactly two points.
- The points of intersection will be symmetrical relative to the y-axis due to the symmetry in the vertex locations.
Now, let's address the given statements:
1. "The second parabola opens downward."
- This is false. If the second parabola also opened downward, they would not intersect at two points.
2. "The second parabola opens upward."
- This is true. For them to intersect at two points, the second parabola must open upward.
3. "The points of intersection are on the x-axis."
- This is false. While it's possible for parabolas to intersect the x-axis, for this specific arrangement relative to their vertices, the points of intersection are not necessarily on the x-axis.
4. "The points of intersection are of equal distance from the y-axis."
- This is true. Given the symmetry around the y-axis, the points where the two parabolas intersect must be equidistant from the y-axis.
### Conclusion:
The correct statements under the given conditions are:
- The second parabola opens upward.
- The points of intersection are of equal distance from the y-axis.
Therefore, the answer to this question is:
- The second parabola opens upward.
- The points of intersection are of equal distance from the y-axis.
