Question

A toddler kicks a ball a distance of 15 feet. The maximum height of the ball above the
ground is 4 feet. If the point at which the child kicks the ball is the origin and the flight of
the ball can be approximated by a parabola, find an expression for the quadratic function
that models the ball's path.

Answer 1

Using x for horizontal distance and y as the height
y = 0 when x = 0 and 15
Let’s guess y = -(x - 0)(x - 15) = -x^2 +15x
Symmetry tells us the maximum height is when x = 7.5 but substituting gives -(7.5)^2 + 15 x 7.5 = 56.25
The equation needs to be flattened by dividing by 56.25 and multiplying by 4

y = -4/56.25 x(x - 15) [ or -16/225 x(x - 15) to avoid the decimal ]

If you don’t want brackets or decimals you get y = -16/225 x^2 + 16/15 x
Check when x = 0 then y = 0
When x = 7.5 then y = -16/225 x (7.5)^2 + 16/15 x 7.5 = 4
When x = 15 then y = -16/225 x 15^2 + 16/15 x 15 = 0