The vertex of a parabola that opens downward is at (0, 4). The vertex of a second parabola is at (0, -4). If the parabolas intersect at two points, which statement must be true?

A. The second parabola opens downward.
B. The second parabola opens upward.
C. The points of intersection are on the x-axis.
D. The points of intersection are of equal distance from the y-axis.

The first parabola is represented as y = a x 2 + 4 where a < 0 .
The second parabola is represented as y = b x 2 − 4 .
Setting the equations equal gives x 2 = a − b − 8 ​ , implying a < b .
The points of intersection are equidistant from the y-axis, so the final answer is $\boxed{The points of intersection are of equal distance from the y-axis.}

Explanation

Understanding the Problem We are given two parabolas. The first parabola opens downward and has a vertex at (0, 4). The second parabola has a vertex at (0, -4). The parabolas intersect at two points. We need to determine which statement must be true.

First Parabola Equation Let's write the general form of the first parabola. Since it opens downward and has a vertex at (0, 4), its equation is of the form y = a ( x − 0 ) 2 + 4 = a x 2 + 4 , where a < 0 .

Second Parabola Equation Now, let's write the general form of the second parabola with a vertex at (0, -4). Its equation is of the form y = b ( x − 0 ) 2 − 4 = b x 2 − 4 , where b can be positive or negative.

Intersection Condition Since the parabolas intersect at two points, we can set their equations equal to each other: a x 2 + 4 = b x 2 − 4 .

Solving for x^2 Rearrange the equation to solve for x 2 : ( a − b ) x 2 = − 8 , so x 2 = a − b − 8 ​ .

Condition for Real x Since x must be real, x 2 must be positive. Therefore, 0"> a − b − 8 ​ > 0 , which implies a − b < 0 , so a < b .

Analyzing b Since a < 0 and a < b , b must be greater than a . If b is positive, the second parabola opens upward. If b is negative, it opens downward, but it must be greater than a .

Analyzing the Options Now let's analyze the given options:

The second parabola opens downward: This is not necessarily true, as b could be positive.

The second parabola opens upward: This is possible, and we need to determine if it must be true.

The points of intersection are on the x-axis: If the points of intersection are on the x-axis, then y = 0 . Substituting into the first equation, 0 = a x 2 + 4 , so x 2 = − 4/ a . Substituting into the second equation, 0 = b x 2 − 4 , so x 2 = 4/ b . Thus, − 4/ a = 4/ b , so b = − a . Since a < 0 , 0"> − a > 0 , so 0"> b > 0 . This is a possible scenario. However, we know that x 2 = − 8/ ( a − b ) . If y = 0 , then x 2 = − 4/ a . Thus − 4/ a = − 8/ ( a − b ) which means 4 ( a − b ) = 8 a or a − b = 2 a or b = − a . This is possible. However, we need to check if this must be true. If b = − a , then x 2 = − 4/ a . So x = ± 2/ − a ​ .

The points of intersection are of equal distance from the y-axis: Since the parabolas are symmetric about the y-axis, and they intersect at two points, the x-coordinates of the intersection points must be opposites of each other. Therefore, the points of intersection are of equal distance from the y-axis.

Intersection Points Since x 2 = a − b − 8 ​ , the x-coordinates are x = ± a − b − 8 ​ ​ . The points of intersection are ( a − b − 8 ​ ​ , a ( a − b − 8 ​ ) + 4 ) and ( − a − b − 8 ​ ​ , a ( a − b − 8 ​ ) + 4 ) . These points are equidistant from the y-axis.

Second Parabola Direction Let's consider if the second parabola must open upward. If it opens downward, then b < 0 . Since a < b , then a < b < 0 . Then a − b < 0 , so 0"> − 8/ ( a − b ) > 0 , which is fine. However, if b < 0 , it is possible to have intersection points. If 0"> b > 0 , then the second parabola opens upward. Since a < 0 , a < b is always true. Therefore, the second parabola does not necessarily open upward.

Conclusion The points of intersection are of equal distance from the y-axis because the parabolas are symmetric with respect to the y-axis.

Final Answer Therefore, the points of intersection are of equal distance from the y-axis.


Examples
Understanding parabolas and their intersections is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile under gravity follows a parabolic path. If we have two projectiles launched from different points with different initial conditions, determining their points of intersection can help predict potential collisions or interactions. This problem illustrates how algebraic analysis of parabolas can provide valuable insights into real-world scenarios involving motion and trajectories.

Answered by GinnyAnswer | 2025-07-07