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.

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

Step-by-step 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

1. 3.4/4

2. 0.8/7

3. 0.5 x 3

4. 0.9 x 6

5. 10.8

6. 2.35

7. 0.567

8. 9.35

9. 0.02

10. 0.06

11. 0.16

12. 0.42

13. 0.63

14. 5.6

15. 8.1

16. 9.9

17. 0.28

18. 0.48

19. 0.27

20. 0.49

21. 0.45

22. 0.72

23. 0.9

24. 0.88

25. 0.75

26. 0.16

27. 0.328

28. 0.495

29. 0.72

30. 1.068