1. 9-4 Quadratic Equations and Projectiles
A model rocket is shot at an angle into the air from the launch pad.
1. The height of the rocket when it has traveled horizontally x feet
from the launch pad is given by h = x² x.
A. Graph this equation
x= -b 2a y x 15 25 35 45 55 70 y
135
184
200
136 1 
200
150
100 50  
x= -(11.43)
2(-.163)   x=
-.326
x 35   x
This is the x coordinate of the vertex

2. When the rocket is 10ft from the pad
B. A 75 ft. tree, 10 ft. from the launch pad, is in the path of the
rocket. Will the rocket clear the top of the tree?
h = -.163(10)² (10)
h =
h = 98
When the rocket is 10ft from the pad
it will be 98 ft. above the ground which
will clear the 75 ft. tree.
Sec. 9-4

3. 2. The rockets height h at t seconds after launch is given by
h = -22.2t² + 133t . -b 2a
-133
2(-22.2)
x =
 3 =
a. Graph this equation.
b. How high is the
rocket in 2 sec.? x 1 2 3 4 5 6 y
 111
 177
 200
110
 -1 y
From the graph
its  175 ft.
Using the
equation its:
200
150
100 50     
h= -22.2t² + 133t
-22.2(2)² + 133•2
177.2 ft.   x
Sec. 9-4

4. General Formula for the Height of a Projectile over Time
Let h be the height (in feet) of a projectile
launched with an initial velocity v feet per second
and an initial height of s feet. Then, after t
seconds, h = -16t² + vt + s .
Since 16ft ≈ 4.9 meters, if the units are in meters
in the formula above, then h = -4.9t² + vt + s .

5. h = -16t² + vt + s h = -16t² + 64t + 10 b. h = -16(3)² + 64(3) + 10
1. A ball is thrown from an initial height of 10 feet with an initial velocity of 64 feet per second.
a. Write an equation describing the height h in feet of the ball after t seconds.
b. How high will the ball be after 3 seconds?
c. What is the maximum height of the ball?
h = -16t² + vt + s
h = -16t² + 64t + 10
b. h = -16(3)² + 64(3) + 10
h =
h = 58 ft
t = - b = -(64) = 2
2a 2(-16)
h = -16(2)² + 64(2) + 10
h =
h = 74 ft

6. h = -4.9t² + vt + s h = -4.9t² + 40 b. h = -4.9t² + 40 0 = -4.9t² + 40
2. An object is dropped from an initial height of 40meters.
a. Write a formula describing the height of the object (in meters)
after t seconds.
b. After how many seconds does the object hit the ground?
c. What is the maximum height of the object?
h = -4.9t² + vt + s
h = -4.9t² + 40
b. h = -4.9t² + 40
0 = -4.9t² + 40
-40/-4.9 = t²
√-40/-4.9 = √t²
√8.2 ≈ √t²
2.9sec. ≈ t
t = - b = = 0
2a 2(-4.9)
h = -4.9(0)² + 40
h = 40m

7. t = - b = -30 ≈ 3 h = -4.9t² + vt + s 2a 2(-4.9) h = -4.9t² + 30t + 10
3. Suppose a ball is thrown upward with an initial upward velocity of 30 meters per second from an initial height of 10 meters.
a Write a formula for the height in meters of the object after t
seconds.
b. Estimate when the ball is 40 meters high.
t = - b = ≈ 3
2a 2(-4.9)
h = -4.9t² + vt + s
h = -4.9t² + 30t + 10
b. 40 = -4.9t² + 30t + 10 80 ◙ ◙
window 0 < x < 10
0 < y < 80
1.2
4.9
1.2 sec and 4.9 sec