Question

The surface of a pedestrian bridge forms a parabola. Let the surface at one side of the bridge be represented by (4,0) and the surface at the other side be represented by (8,0). The center of the bridge is 3 feet higher than each side and can be represented by a vertex of (6,3). Write a function in vertex form that models the surface of the bridge.

Answer 1

y = -3/4(x - 6)^2 + 3

Explanation:

y = ax^2 + bx + c

(4, 0)

16a + 4b + c = 0 Eq. 1

(8, 0)

64a + 8b + c = 0 Eq. 2

(6, 3)

36a + 6b + c = 3 Eq. 3

We have a system of three equations in three variables.

Eq. 2 - Eq. 1

48a + 4b = 0 Eq. 4

Eq. 2 - Eq. 3

28a + 2b = -3 Eq. 5

Eq. 4 - 2 * Eq. 5

-8a = 6

a = -6/8

a = -3/4

48a + 4b = 0

48(-3/4) + 4b = 0

4b = 36

b = 9

16a + 4b + c = 0

16(-3/4) + 4(9) + c = 0

-12 + 36 + c = 0

c = -24

The equation of the parabola is

y = -3/4 x^2 + 9x - 24

y = -3/4(x^2 - 12x + 32)

y = -3/4(x^2 - 12x + 36) + (-3/4)32 - (-3/4)(36)

y = -3/4(x^2 - 12x + 36) -24 + 27

y = -3/4(x - 6)^2 + 3