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Find the equation of a sine curve whose peak is (2, 4) and valley is (10, -6).

User Ubeogesh
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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:

  • T = (10 - 2) * 2 = 16

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:

  • Phase shift = -2

Putting it all together, the equation of our sine curve is:

y = 5sin(π/8(x + 2)) - 1

User Vkelman
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