Final answer:
A parabola that opens down has a quadratic function with a negative leading coefficient for the x² term, indicating that 'a' is less than zero (a < 0). This reflects the shape of various physical phenomena depicted by parabolic curves, such as projectile motion or potential energy in oscillators.
Step-by-step explanation:
If a parabola opens down, this means that the quadratic function representing the parabola will have a negative coefficient for the x² term. In the standard form y = ax² + bx + c, 'a' would be less than zero (a < 0). This is because the sign of the leading coefficient 'a' determines the direction in which the parabola opens. A negative 'a' results in a parabola that opens downwards, with its vertex representing the maximum point. Conversely, a positive 'a' would make the parabola open upwards, and the vertex would then represent the minimum point.
When translating this into physical situations, such as the trajectory of a ball thrown in the air or the potential energy of an oscillator, a downward-opening parabola can represent different things. For instance, in projectile motion, a downward opening parabola can represent the path of a projectile under the influence of gravity. Meanwhile, in the context of oscillators, it might represent the potential energy of the system which is a quadratic relationship with the position.