Final answer:
The equation of the sine curve with a peak at (2, 4) and a valley at (10, -6) is y = 5sin(π/8(x + 2)) - 1, involving an amplitude of 5, a vertical translation of -1, a period of 16, and a phase shift of -2.
Step-by-step explanation:
The question is asking to find the equation of a sine curve given its peak and valley points. The peak is at (2, 4) and the valley is at (10, -6).
First, we determine the amplitude of the sine function. The amplitude is half the distance between the peak and the valley, so in this case:
- Amplitude = (4 - (-6)) / 2 = 10 / 2 = 5
Next, we find the vertical translation, which is the midpoint between the peak and the valley:
- Vertical translation = (4 + (-6)) / 2 = -1
The period (T) of the sine function is the distance between the peak and the valley along the x-axis times two:
Now we can get the b-value using the formula for the period T = 2π / b:
- b = 2π / T = 2π / 16 = π / 8
Since the sine curve starts at the peak at x = 2 instead of the traditional 0, there is a phase shift to account for:
Putting it all together, the equation of our sine curve is:
y = 5sin(π/8(x + 2)) - 1