Given the equation y=3sin(3x-180)-2 state the domain, range, amplitude, period, and phase shift of the cycle.
as you know,
a sin(k(x-h))+b
has
domain all reals
amplitude a
period 2pi/k
shift h
range [b-a,b+a]
so, put yours into that form and read off the values
To determine the domain, range, amplitude, period, and phase shift of the given equation, let's break it down:
1. Domain:
The domain of a sine function is all real numbers since the sine function is defined for any input value.
Domain: All real numbers
2. Range:
The range of a sine function is typically between -1 and 1, but since this equation has been adjusted, we need to determine the new range.
In this equation, the coefficient of sin(3x-180) is 3, and the constant -2 is subtracted from the whole function. Therefore, the range is shifted downward by 2 units and multiplied by 3 units.
Range: [-2-3, -2+3] = [-5, 1]
3. Amplitude:
The amplitude of a sine function is the absolute value of the coefficient in front of the sin(). In this case, the coefficient is 3.
Amplitude: 3
4. Period:
The period of a sine function is given by 2π divided by the coefficient inside the sin().
Period: 2π/3
5. Phase Shift:
To determine the phase shift, we need to isolate the value inside the sin() function. In this case, it is (3x - 180).
To find the phase shift, we set the expression inside the parentheses equal to zero and solve for x:
3x - 180 = 0
3x = 180
x = 60
The phase shift is the value of x (in degrees) that results in sin(0). Since sin(0) = 0, the phase shift in this case is 60 degrees to the right.
Phase Shift: 60 degrees to the right
Summary:
Domain: All real numbers
Range: [-5, 1]
Amplitude: 3
Period: 2π/3
Phase Shift: 60 degrees to the right