Answer:
- amplitude 3
- period 2π/3
- phase shift π/2
- range -5 to +1
- y-intercept +1
Explanation:
You want the amplitude, period, phase shift, range, and y-intercept of the function ...
f(x) = 3sin(3x -3π/2) -2
Transformed function
The amplitude of the parent sine function is 1. The period of it is 2π. When the function has a different amplitude and period, it looks like ...
f(x) = (amplitude)·sin(2πx/(period))
Comparing this form to ...
f(x) = 3sin(3x)
we find the multiplier is 3 and the argument of the sine function is 3x. This tells us ...
amplitude = 3
(2πx/period) = 3x ⇒ period = (2πx)/(3x) = 2π/3
Translation
When a function is translated h units to the right, and k units up, it becomes ...
f(x -h) +k
Looking at the attached graph of the given function, we see the point that is (0, 0) on the parent function is translated to (π/2, -2) on the graph. Then our translated function is ...
f(x -π/2) -2 = 3sin(3(x -π/2)) -2
The amount of horizontal translation is the phase shift, which is π/2.
Range
The range of the function is the interval between (and including) the maximum and minimum. Since the amplitude is 3 and the vertical shift is -2, the range is -2±3 = -5 to +1. In interval notation, the range is [-5, 1].
Y-intercept
The y-intercept is the value of y when x = 0. For the given function, that is ...
f(0) = 3·sin(3·0 -3π/2) -2 = 3·sin(-3π/2) -2 = 3 -2 = 1
The y-intercept is 1.
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