To determine the correct equation of a sine function with the specified range and period, let's analyze the requirements.
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Range: The range is given as \( 6 \leq y \leq 9 \). This means:
- The midline (average of maximum and minimum) is \( (9 + 6) / 2 = 7.5 \).
- The amplitude is \( (9 - 6) / 2 = 1.5 \).
Therefore, the general form of the sine function should be: \[ y = 1.5 \sin(kx) + 7.5 \] where \( k \) affects the period.
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Period: The period of the sine function is determined by the formula: \[ \text{Period} = \frac{360°}{k} \] We want a period of \( 60° \). Setting the equation for the period: \[ 60° = \frac{360°}{k} \implies k = \frac{360°}{60°} = 6 \]
Combining these elements, the sine function that meets both the range and period requirements is: \[ y = 1.5 \sin(6x) + 7.5 \]
Now, let’s evaluate the options:
a) \( y = \sin(6x) + 9 \) - Range is \( 8 \leq y \leq 10 \) (incorrect). b) \( y = 3 \sin(12x) + 9 \) - Range is \( 6 \leq y \leq 12 \) (incorrect). c) \( y = 3 \sin(12x) + 6 \) - Range is \( 3 \leq y \leq 9 \) (incorrect). d) \( y = 1.5 \sin(6x) + 7.5 \) - Range is \( 6 \leq y \leq 9 \) (correct).
The correct choice is: d) \( y = 1.5 \sin(6x) + 7.5 \).