A sine function has the following key features:
Period = 4
Amplitude = 3
Mid-line: y=−1
y-intercept: (0, -1)
The function is not a reflection of its parent function over the x-axis.
The first point must be on the mid-line and the second point must be a maximum or minimum value on the graph closest to the first point.
so, after all the others whose solution you have seen, what bothers you about this one?
To determine the equation of the sine function that satisfies the given conditions, we can start by using the general form of a sine function:
y = A * sin(Bx + C) + D
where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.
From the given information:
Amplitude = 3
The amplitude is the distance from the mid-line to the maximum or minimum value of the graph. In this case, the mid-line is y = -1, so the maximum value would be 2 units above the mid-line and the minimum value would be 2 units below the mid-line. Therefore, the amplitude is 3.
Period = 4
The period is the distance it takes for one complete cycle of the graph. In this case, the period is 4 units.
Since the sine function has a period of 2π, we can use the formula:
B = 2π / period
In this case, B = 2π / 4 = π/2
Mid-line: y = -1
The mid-line is the horizontal line that the graph oscillates around. In this case, the mid-line is y = -1.
y-intercept: (0, -1)
The y-intercept is the point where the graph intersects the y-axis. In this case, the y-intercept is (0, -1).
The function is not a reflection of its parent function over the x-axis.
Since the function is not a reflection over the x-axis, the phase shift (C) remains 0. If it were reflected, the phase shift would be π.
First point on the mid-line and second point must be a maximum or minimum value closest to the first point.
The first point should be on the mid-line, which is y = -1. The second point should be either a maximum or minimum value that is closest to the first point.
Putting all the information together, the equation of the sine function that satisfies the given conditions is:
y = 3 * sin((π/2)x) - 1