3.5 QUARDRATIC MODELS USING FACTORED FORM
Learning Goal: We will write the equation of a quadratic model using the factored form of a
quadratic relation.
The quadratic relation in factored form can be used to model a curve of best fit for a set of data
that approximates the shape of a parabola and passes through the horizontal axis.
The actual or estimated x-intercepts or zeros of a curve of best fit represent the values of r and
s in the quadratic relation in factored form y = a(x – r)(x – s).
The value of a of the quadratic relation in factored form y = a(x – r)(x – s) can be determined by
substituting the value of an ordered pair located on or near the curve of best fit.
Examples:
1. Data collected from the flight of a golf ball are shown below.
Horizontal Distance (m)
Height (m)
0
0.0
20
22.0
40
29.8
50
27.0
60
22.5
(a) Determine the value of the zeros.
(b) Determine an equation for a curve of best fit.
(c) Use the equation to determine the height of the golf ball when the horizontal
distance of the golf ball is 30 m.
2. The following data describes the flight of a glider launched from a tower on a
hilltop. The height values are negative whenever, the glider was below the
height of the hilltop.
Time
(s)
Height
(m)
0
1
2
3
4
5
6
7
8
9
10
4
1.75
0
-1.25
-2
-2.25
-2
-1.25
0
1.75
4
(a) Graph the data.
(b) How tall is the tower?
(c) Find an equation to model the
flight of the glider.
(d) Find the minimum point in the glider’s flight.
3. Craig and Ben are analyzing data collected from a motion detector following
the launch of their model rocket.
Time (s)
Height (m)
0.0
0.0
1.0
16.0
2.0
20.0
3.0
15.5
4.0
0.0
(a) Determine an equation for a curve of good fit.
(b) Use the equation to estimate the height of the rocket 0.5 s after it is launched.
4. A football is kicked into the air. Its height above the ground is approximated
by the relation h = -5t (t – 4), where h is the height in metres and t is the time
in seconds since the football was kicked.
(a) When does the football hit the ground?
(b) After how many seconds does the football reach its maximum height? What is
the maximum height?
5. Grace hits a golf ball that follows a parabolic path. The ball lands 120 m from
where it was struck. The ball was 40 m above the ground when it was 20 m
short of the hole.
(a) Draw a diagram that models the flight of the ball.
(b) Determine an equation that models the flight of the ball.
(c) What was the maximum height of the golf ball?
Extra Practice – Quadratics Application Problems
1. The Rainbow Bridge in Utah is a natural arch that is approximately parabolic in shape.
The arch is about 88 m high. It is 84 metres across at its base. Determine a quadratic
equation in factored form; that models the shape of the arch.
2.
A king fisher dives into a lake. The underwater path of the bird is described by a
parabola with the equation y = 0.5x2 – 3x, where x is the horizontal position of the bird
relative to its entry point and y is the depth of the bird underwater. Both
measurements are in metres. How far does the bird swim underwater? What is the
bird’s greatest depth below the water surface? [Hint: Create a table of values for x = 1 to x = 6]
3. A rock is launched into the air from a catapult and follows a parabolic path. If the rock
hits the ground after 9 seconds, and the maximum height of the ball is 6.75 metres, then
what is the height of the ball after 3 seconds?
4. The opening under a bridge is parabolic in shape. The opening is 28 m wide at the
bottom and it is 52.5 m high at a point 7 m to the right of the centre. Find the maximum
height of the opening.
5. A soccer ball is kicked from a point and lands 40 m away. It reaches a maximum height
of 10 m during its parabolic flight.
(a) Draw a sketch, label the axis and three points (as ordered pairs)
(b) Determine the equation, in factored form, to model the height of the soccer ball, h,
in terms of distance travelled, d. Assume both dimensions are measured in metres.
6. A rocket is launched into the air from the ground. The height of the rocket, h, is given by
the equation h = -5t2 + 60t, where t is the time after launch, in seconds, and h is the
height above ground, in metres.
(a) Find how long the rocket is in the air.
(b) Determine at what time the rocket will reach its maximum height.
(c) Find the maximum height the rocket attains.
(d) How high is the rocket after 3 seconds?
7. Grace hits a golf ball out of a sand trap, from a position that is level with the green. The
path of the ball is approximated by the equation y = -x2 + 5x, where x represent the
horizontal distance travelled by the ball in metres and y represents the height of the ball
in metres. Determine the greatest height of the ball and the distance away that it lands.
[Hint: Create a table of values]
Answers:
1. h =
w (w – 84) OR h =
(w – 42) (w + 42)
3. 6 metres
5. h = d (d – 40)
6. (a) 12 seconds (b) 6 seconds (c) 180 metres (d) 135 metres
2. 6 metres, 4.5 metres
4. 70 metres
7. 6.25 metres, 5 metres