Question 1
The period of \( y = \sin x \) (in degrees) is:
Solution:
The parent sine function has period 360°.
Question 2
The amplitude of \( y = 5\sin x \) is:
Solution:
Amplitude = \( |a| = 5 \).
Question 3
For \( y = \sin x \), the value at \( x = 90° \) is:
Solution:
\( \sin 90° = 1 \).
Question 4
The range of \( y = \cos x \) is:
Solution:
Cosine oscillates between -1 and 1.
Question 5
For \( y = 3\sin(2x) + 1 \), the period (degrees) is:
Solution:
Period = \( \frac{360°}{|k|} = \frac{360°}{2} = 180° \).
Question 6
For \( y = 4\sin(x) + 2 \), the maximum value is:
Solution:
Max = midline + amplitude = 2 + 4 = 6.
Question 7
For \( y = -2\cos(x) - 5 \), the minimum value is:
Solution:
Amplitude 2, midline \( y = -5 \). Min = -5 - 2 = -7.
Question 8
For \( y = \sin(x - 30°) \), the phase shift is:
Solution:
\( y = \sin(x - d) \) shifts horizontally right by \( d \). Right 30°.
Question 9
A Ferris wheel has diameter 30 m. The lowest point is 2 m above ground. The midline (axis of the wheel) is at height (m):
Solution:
Radius = 15. Midline = 2 + 15 = 17 m.
Question 10
A high tide of 8 m occurs every 12 hours. Low tide is 2 m. Amplitude (m):
Solution:
Amplitude = (max - min)/2 = (8 - 2)/2 = 3 m.
Question 11
From Q10, the equation \( h(t) = a\cos(k(t - d)) + c \) with peak at \( t = 0 \) has \( c = ? \)
Solution:
\( c \) = midline = (max + min)/2 = (8 + 2)/2 = 5 m.
Question 12
For a tide with period 12 h, the value of \( k \) (in degrees per hour) is:
Solution:
\( k = \frac{360°}{12} = 30°/\text{h} \).