Question

Question 11 (1 point) ~ Saved
Determine the equation of a sine function that would have a range of (ye R6 sy≤9) and a period of 60°.
a) y = sin(6x) + 9
b)
y = 3 sin(12x) + 9
C)
y = 3 sn(12x) + 6
d) y = 1.5 sin(6x) + 7.5

Answer 1

To determine the correct equation of a sine function with the specified range and period, let's analyze the requirements.

1. **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.

2. **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) \( y = 1.5 \sin(6x) + 7.5 \)**.