An inverted parabola (arch) has feet 10 metres apart and its highest point is 10 metres. Find the equation of the parabola.

Question

 the arch is a parabolic arch in a shape of an inverted parabola

Answer 1

As Mark says there is more than one solution to this problem depending on where we place the feet of the arch on the x-axis.

 

Using the standard placement with feet at -5 and at 5, we know the parabola has two roots, hence factors as

 

y = d(x - 5)(x + 5) = d(x²-25)

 

We know the vertex is at -b/2a, or at 0 (halfway between the two feet) (we've already determined that b=0, so -b/2a = 0), so the vertex is at (0,10)

 

10 = d(0²-25)

d = -10/25 = -2/5

 

Altogether,

y = (-2/5)x² + 10

Answer 2

There is more than one possible answer to this question, depending on the coordinate system used.  One possibility would be to use the line through the "feet" as the x-axis with the origin halfway between the feet.  In that case, the feet have coordinates (-5,0) and (5,0) and the highest point of the arch has coordinates (0,10).  

 

An equation of the parabola is then y = a(x-0)²+10.

 

Since (5,0) is on the graph, 0 = 25a+10

 

                                      25a = -10

 

                                         a = -2/5

 

Equation of parabola:  y = (-2/5)x² + 10