Answer: (a) Amplitude = 19
(b) Vertical Shift = 59
(c) Period = 24
[tex]\bold{(d)\quad y=19\cos\bigg(\dfrac{\pi}{12}x\bigg)+59}[/tex]
(e) The model = 50°, the actual value = 52°
Step-by-step explanation:
The equation of a cosine function is: y = A cos (Bx - C) + D where
- Amplitude (A) is the distance from the center to the maximum
- Period (P) = 2π/B --> B = 2π/P
- Phase Shift = C/B
- Center (D) is the vertical shift
(a) Amplitude (A) = (Max - Min)/2
= (77 - 40)/2
= 37/2
= 18.5 (I rounded it up to 19)
(b) Vertical Shift (aka Center) (D)= (Max + Min)/2
= (77 + 40)/2
= 117/2
= 58.5 (I rounded it up to 59)
(c) Period (P) = 24
B = 2π/P
= 2π/24
= π/12
(d) There is no Phase Shift because the max is on the y-axis.
A = 19, B = π/12, C = 0, D = 59
⇒ y = 19 cos(π/12)x + 59
(e) let x = -2 (-2 represents 10 am on the graph)
⇒ y = 19 cos(π/12)(-2) + 59
= 19 cos (-π/6) + 59
= 19(-1/2) + 59
= -9.5 + 59
= 49.5
The model estimates the temperature at 10 am to be ≈ 50°
The actual temperature from the table is 52°
