QUADRATIC MODELS USING FACTORED FORM Learning Goal: We will write the equation of a quadratic model using the factored form of a quadratic relation. Example:

1. Data collected from the flight of a golf ball are shown below.

Horizontal Distance (m) 0 20 40 50 60

Height (m) 0.0 22.0 29.8 27.0 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)

0 1 2 3 4 5 6 7 8 9 10

Height (m)

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.

CLASS WORK – Try these on your own! If you are stuck, make sure to ask for help! 3. Craig and Ben are analyzing data collected from a motion detector following the launch of

their model rocket.

Time (s) 0.0 1.0 2.0 3.0 4.0 Height (m) 0.0 16.0 20.0 15.5 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?

Solutions: 3a) y = -5 (x)(x – 4), b) 8.76 m 4a) after 4 seconds (zeroes), b) 2 seconds, 20 m 5b) y = -0.02x(x – 120), c) 72 m Homework: Page 203 #8 Page 205 # 7

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 = 0.5x(x – 3), 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?

3. A rock is launched into the air and follows a parabolic path. If the rock hits the ground

after 9 seconds, and the maximum height of the rock is 6.75 metres, then what is the height of the rock 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 = -5t(t - 12), 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?

Answers:

1. h =

x (x – 84) OR h =

(x – 42) (x + 42)

2. 3 m, 1.125 m under the water 3. 6 m 4. 70 m

5. h =

d (d – 40)

6. (a) 12 seconds (b) 6 seconds (c) 180 metres (d) 135 metres