Parabola and quadratic function word problem

Question

The shape of a suspension bridge can be turned upside down to create a bridge where each supporting truss is in the shape of a parabola. The truss of the bridge shown in the figure is 45 feet high at its highest point, and the two points where it touches the ground are 60 feet apart, as shown in the figure. Find a quadratic function that models the parabolic truss.

Answer 1

Think that the highest point is on the vertical axis. Its coordinates are (0, 45).

The two points at which it touches the ground are (-30, 0) and (30, 0).

The zeros of the function are ±30.

 

f(x) = ax² + bx + c

 

45 = a(0)² + 0b + c

45 = c

 

0 = a(30)² + 30b + 45

0 = 900a + 30b + 45

 

and

 

0 = a(30)² - 30b + 45
0 = 900a - 30b + 45

 

Using elimination by adding

 

0 = 1800a + 90

-90 = 1800a

-1/20 = a

 

Substituting into either of the two above

 

0 = -(30)²/20 + 30b + 45

0 = -900/20 + 30b + 45

0 = -45 + +30b + 45

0 = b

 

f(x) = -x²/20 + 45