Explanation:
if you need to know the formula yourself, then this is actually physics and not just mathematics.
anyway, we assume this happens on the surface of Earth.
so, with Earth's gravity and such the formula for the height of the bullet at a certain time is
h(t) = -16t² + v0×t + staying height =
= -16t² + 1200t + 5
the x-intercept is the time, when the bullet falls back down and hits the ground. in other words. when it's height = 0.
0 = -16t² + 1200t + 5
a quadratic equation
ax² + bx + c = 0
has the genetic solution
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = t
a = -16
b = 1200
c = 5
t = (-1200 ± sqrt(1200² - 4×-16×5))/(2×-16) =
= (-1200 ± sqrt(1,440,000 + 320))/-32 =
= (-1200 ± sqrt(1,440,320))/-32 =
= (-1200 ± sqrt(64×22505))/-32 =
= (-16×75 ± 8×sqrt(22505))/-32 =
= (75/2 ± 1/4 × sqrt(22505))
t1 = 75/2 + 1/4 × 150.0166657... = 75.00416644... seconds
t2 = 75/2 - 1/4 × 150.0166657... = -0.004166435... seconds
after about 75 seconds the bullet will hit the ground (x-intercept).
the t2 solution is the theoretical extension of the flight curve backwards in time (remember, the staying height is 5 ft, but using the shot graph and the corresponding accelerations the bullet would have started 0.004166435... seconds before the shot on the ground, and this is therefore also an x-intercept.
for the vertex (the max. height of the bullet) we need the zero of the first derivation (the extreme point of the function).
h'(t) = -32t + 1200 = 0
32t = 1200
t = 1200/32 = 37.5 seconds
so, after 37.5 seconds the bullet reaches is maximum height, which is
-16×37.5² + 1200×37.5 + 5 = 22,505 ft
so, the vertex is (37.5, 22505).