- The amplitude is 8.
- The period is 2π/61.
- The horizontal shift is 42 units to the right.
- The midline is y = 2.
The equation y = 8 sin(61x - 42) + 2 represents a sinusoidal function. Let's analyze the equation to determine the amplitude, period, horizontal shift, and midline.
1. Amplitude:
The amplitude of a sinusoidal function is the maximum absolute value of the function's range. In this case, the amplitude is 8. The coefficient in front of the sine function, which is 8, determines the amplitude. The amplitude represents the vertical distance from the midline to the peak or trough of the function.
2. Period:
The period of a sinusoidal function is the horizontal length of one complete cycle of the function. To find the period, we use the formula T = 2π/|b|, where b is the coefficient of x in the argument of the sine function. In this case, the coefficient is 61. Therefore, the period is T = 2π/61.
3. Horizontal Shift:
The horizontal shift of a sinusoidal function is the amount by which the function is shifted to the left or right. To find the horizontal shift, we set the argument of the sine function, 61x - 42, equal to zero and solve for x. In this case, x - 42 = 0, which gives us x = 42. The horizontal shift is 42 units to the right.
4. Midline:
The midline of a sinusoidal function is the horizontal line around which the function oscillates. In this case, the midline is y = 2. The midline represents the average or equilibrium value of the function.