A sine function has the following key features:
Period = π
Amplitude = 2
Mid-line: y=−2
y-intercept: (0, -2)
The function is 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.
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To identify the first point on the mid-line, we know the y-coordinate is -2, as given in the key features.
First Point: (0, -2)
Next, we need to find the maximum or minimum value on the graph closest to the first point. Since the sine function is a reflection of its parent function over the x-axis, the first point will be a minimum value.
Knowing the period of the sine function is π, the length of one full oscillation is equal to π.
The second point that is a minimum value closest to the first point will be halfway through one full oscillation. Therefore, it will be located at half the period, which is π/2.
To find the y-coordinate of this point, we can use the amplitude, which is 2, and the mid-line, which is -2.
Since the amplitude determines how far above and below the mid-line the function reaches, we can add the amplitude to the mid-line to find the maximum value and subtract the amplitude to find the minimum value.
Maximum Value: -2 + 2 = 0
Minimum Value: -2 - 2 = -4
The second point, which is a minimum value, will have a y-coordinate of -4.
Second Point: (π/2, -4)
Therefore, the two points on the sine function graph with the given key features are:
First Point: (0, -2)
Second Point: (π/2, -4)
To find the equation of the sine function with the given key features, we can start by using the standard form of the sine function:
y = A * sin(Bx - C) + D
where:
A is the amplitude,
B is the frequency,
C is the phase shift, and
D is the vertical shift.
Let's use the given information to find the values of A, B, C, and D.
Amplitude:
The given amplitude is 2. So, A = 2.
Period:
The period of the sine function is given as π. The period is the distance between two consecutive peaks (or troughs) of the function. In the standard form, the period is calculated as 2π/B. We can equate this to the given period and solve for B:
2π/B = π
B = 2
So, B = 2.
Mid-line:
The mid-line represents the vertical shift of the graph. The given mid-line equation is y = -2. The value of D is the same as the mid-line value. Therefore, D = -2.
Phase Shift:
The function is a reflection of its parent function over the x-axis, which means the phase shift is π units to the right. This is because a reflection about the x-axis causes a shift in the opposite direction along the x-axis. So, C = π.
Now that we have the values of A, B, C, and D, we can substitute them into the standard form of the sine function:
y = 2 * sin(2x - π) - 2
This is the equation of the sine function with the given key features.
To find the first point on the mid-line, we substitute x = 0 into the equation:
y = 2 * sin(2(0) - π) - 2
y = 2 * sin(-π) - 2
y = 2 * 0 - 2
y = -2
So, the first point on the mid-line is (0,-2).
To find the second point, we need to identify the maximum or minimum value closest to the first point. Since the amplitude is positive, the point closest to the first point will be a maximum value. In a sine function, maximum values occur halfway between consecutive troughs. In other words, we need to find the x-coordinate where the sine function has a maximum value.
The general equation for the x-coordinate of a maximum value in a sine function is given as:
x = (n * π) + (π/2)
where n is an integer representing the number of complete cycles. Since the period of the function is π, there is exactly one complete cycle in each interval of π. Therefore, for the value of x closest to 0, n = 0.
x = (0 * π) + (π/2)
x = π/2
Substituting this value of x into the equation:
y = 2 * sin(2(π/2) - π) - 2
y = 2 * sin(π - π) - 2
y = 2 * sin(0) - 2
y = 2 * 0 - 2
y = -2
So, the second point is (π/2, -2).
Therefore, the two points on the graph of the given sine function are (0,-2) and (π/2, -2).