Seminar 3 Application Activity: Angry Birds Consider the following scenario: Red
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Question
Seminar 3 Application Activity: Angry Birds
Consider the following scenario:
Red Bird, Yellow Bird, Blue Bird and Black Bird are angry with the pigs who stole the birds’ eggs. The birds want their eggs back and will stop at nothing to get them back. The flight path of the birds can be modeled with a parabola where “x” is the distance and “y” is the height.
Use the data below to help answer the following questions:
Red Bird starts his flight from point (10, 0). His flight path reaches a maximum height of 18 yards and lands at point (38, 0).
Yellow Bird’s flight path can be modeled by the quadratic equation y=-X^2+14x-24
Blue Bird’s flight is modeled by the following graph:
The table below contains partial data points of Black Bird’s trajectory:
x
8
9
10
11
12
13
14
15
16
17
18
y
0
7.5
14
19.5
24
27.5
30
31.5
32
31.5
In developing responses to the problems, be sure to show all work:
What is the maximum height of each bird’s flight: (4points)
What is the axis of symmetry for each bird’s flight: (4 points)
What was the total horizontal distance of each bird’s flight: (4 points)
Which bird flew the highest? (2 points)
Which bird traveled the greatest horizontal distance? (2 points)
Which bird hit the following pigs:
King Pig located at point (21, 19.5) (2 points)
Moustache Pig located at point (9, 21) (2 points)
x
8
9
10
11
12
13
14
15
16
17
18
y
0
7.5
14
19.5
24
27.5
30
31.5
32
31.5
Explanation / Answer
The vertex form of the equation of a parabola is y = a(x –h)2 + k, where (h ,k) is the vertex.
The points (10, 0) and (38, 0) are on the Red Bird’s trajectory. On substituting x = 10 and y = 0 in the above quadratic equation, we get 0 = a(10 -h)2 + k ..(1). Similarly, on substituting x = 38 and y = 0 in the above quadratic equation, we get 0 = a( 38- h)2 + k… (2). Further, since a parabola is a symmetric figure (in this case, it is symmetric about a line parallel to y-axis), the x-coordinate of the vertex is 10 + (38 – 10)/2 i.e. 10 + 28/2 or, 10 + 14 = 24 and the y-coordinate is the maximum height of the flight path, i.e. 18. Hence h = 24 and k = 18. On substituting these figures in the equations (1) and (2) above, we get 0 = a(10 -24)2 + 18 and 0 = a(38-24)2 + 18. Then, a(±14)2 +18 = 0 or 196a = -18 so that a = -18/196 = -9/98. Then the flight path of the Red Bird is given by y = - 9/98(x – 24)2 + 18…(RB).
The Yellow Bird’s flight path is given by the quadratic equation y= - x2 +14x -24 = -( x2 - 2*7x + 49) – 24 + 49 = - (x – 7)2 + 25 i.e. y =- (x – 7)2 + 25 …(YB)
The graph is not available so that we may not determine the flight path of the Blue Bird.
From the data points given for the Black Bird’s flight path, it is apparent that the coordinates of the vertex are (16, 32) as it achieves a maximum height of 32 when x = 16. Let the equation of its flight path be y = a(x-16)2 + 32. On substituting x = 8 and y = 0 in this equation, we get 0 = a(8 –16)2 + 32 or, a( -8)2 + 32 = 0 or, 64a = -32 so that a = -32/64 = -1/2. Then the equation for the Black Bird’s flight path is y = -1/2(x -16)2 + 32…(BB)
Red Bird ------ x = 24
Yellow Bird--- x = 7
Black Bird ----- x = 16 ( all of these are the vertical lines passing through the respective vertices)
Blue Bird --- cannot be determined
3. The total horizontal distance of the birds’ flights are as under:
Red Bird ------ 38-10 = 28 ( these are the x –coordinates of the starting point and the end-point of the flight path where y = 0)
Yellow Bird--- On substituting y = 0 in the equation YB, we get - (x – 7)2 + 25 = 0 or, (x -7)2 = 25 so that (x -7) = ± 25 i.e. either x-7 = + 25 or – 25. Then x = 25+7 = 32 or x = -25 + 7 = -18. Thus, the total horizontal distance of the Yellow birds’ flight is 32 + 18 = 50.
Black Bird ---- From the data points given, we observe that the axis of symmetry of the Black Bird’s flight path is x = 16( this line passes through the vertex). The x-coordinate of starting point being 8, the horizontal distance travelled by the Black Bird is 2 (16-8) = 2*8 = 16.
This distance for the Blue Bird’s flight cannot be determined.
4. The y-coordinates of the vertex of the Red Bird, Yellow Bird and the Black Bird are 18, 25 and 32 respectively. Thus, of these 3 birds, the Black Bird flew the highest. Nothing can be said about the Blue Bird.
5. The maximum horizontal distance travelled amongst the Red Bird, Yellow Bird and Black Bird is by the Yellow Bird (50).
6. The King Pig is located at point (21, 19.5). It cannot be on the Red Bird’s flight path as its maximum height is 18.On substituting x = 21 and y = 19.5 in the flight path of the Yellow Bird, we have 19.5 = - (21 – 7)2 + 25 or, 19.5 = 142 + 25 which is not correct. Hence the King Pig is not hit by the Yellow Bird. On substituting x = 21 and y = 19.5 in the flight path of the Black Bird, we have 19.5 = -1/2(21 -16)2 + 32 or, 19.5 = -1/2(25) + 32 = -12.5 + 32 which is correct. Therefore, the Black Bird hits the King pig.
7. The Moustache Pig is located at point (9, 21). It cannot be on the Red Bird’s flight path as its maximum height is 18.It cannot be on the Black Bird’s flight path either since as per the data points given y = 7.5 when x = 9. On substituting x = 9 and y = 21 in the flight path of the Yellow Bird, we have 21 =- (9 – 7)2 + 25 or, 21= 4 + 25 which not correct. Hence the Moustache pig is not on the flight path of Yellow Bird also. We cannot say anything about the Blue Bird.
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