Question

A building has an entry the shape of a parabolic arch 84 ft high and 42 ft wide at the base, as shown below. A parabola opening down with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 84 feet and its width from left to right is 42 feet. Find an equation for the parabola if the vertex is put at the origin of the coordinate system. (1 point)

Answer 1

Since the parabola passes by the center its equation is:

y = ax². But it opens downward, that means the coefficient a is negative.

Then the equation becomes:

y = - ax², with x = 0 as its axis of symmetry.

We are given that the height is 84 ft when the opening downward is 42 ft.
That means to the (height) y, corresponds x =+21 & x=-21 (due to symmetry).
In order to calculate a let's plug y & x with their related values:

y = - ax²

84 = - a(21)²

84 = - a(441) and a = - 84/441 ↔ a = - 4/21

 And the final equation is : y = -4/21. x²