A marching band performs on the football field at half-time. As they perform, the members of the bandstand in the shape of a sinusoidal function. While playing, they move, but still maintain the sinusoidal function, transforming it in different ways.
Darla is a member of the marching band. As the band begins to play she is positioned in the exact center of the field. The person closest to her on the same horizontal line stands 10 yards away. The sinusoidal function extends to the ends of the playing field.
The playing area of the football field measures 300 feet by 160 feet. Place the playing area of a football field on the coordinate plane such that the origin is the lower-left corner of the football field.
1. Edna is sitting in the stands and is facing Darla. Edna observes that sine curve begins by increasing at the far left of the field. What is the equation of the sine function representing the position of band members as they begin to play?
y = A sin (Bx - C) + D
A: amplitude = max - D or D - min
B: Period = 2π/B ---> B = 2π/Period
C: Phase Shift = C/B ---> C = B · Phase Shift
D = 80 the field has a height of 160 ft so the center is at y = 80
A = 65 the min is at 5 yds (15 ft). D - min = 80 - 15 = 65
B = 60 the person next to Darla is 10 yds (30 ft) --> Period = 60
B = 2π/60 = π/30
C = 5π the field is 300 ft wide so the center is 150 ft
Phase Shift = 150. C = B · Phase Shift = π/30 · 150 = 5π
A = (-) the band ends down (at 15 feet) so A is negative