UNO High School Problem of the Week Competition
Problem 7
A parabola passes through the points (2,0) on the x-axis and (0,1) on the y axis. Also it opens downwards and has its vertex on the line y = 2. If its vertex is at (a, 2), what is the value of a?
Solution
A parabola that opens downward with a vertex at (a,2) will have the equation
y-2 = -b(x-a)2
for some constant b.
By plugging in our two points, (0, 1) and (2,0) we can get two equations:
1-2 = -b(0-a)2 or 1 = ba2 and
0-2 = -b(2-a)2 or 2 = b(2-a)2.
By using the first equation we find that b = 1/a2 and can substitute this into the second equation to get:
2 = (1/a2(2-a)2
2a2 = (2-a)2 = 4 - 4a +a2
a2 + 4a - 4 = 0
We can now use the quadratic formula to find that a = 2(squareroot(2)-1) or a = 2(-squareroot(2)-1) ;
