Answer: C. f(t) = -3sin(3t)
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Work Shown:
We can immediately rule out choice B because plugging t = 0 into f(t) = 6*cos(3t) yields the following
f(t) = 6*cos(3t)
f(0) = 6*cos(3*0)
f(0) = 6*cos(0)
f(0) = 6*1
f(0) = 6
But the graph does not go through (0,6) and instead it goes through (0,0).
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So we know for sure its some sine function.
General form of sine
y = A*sin(B(x-C))+D
|A| = amplitude
B = 2pi/T, with T being the period
C = phase shift
D = midline
The graph shows a period of 2pi/3. This is the red portion of the graph which constitutes one full cycle. The graph repeats this red portion infinitely in both directions.
Note: You don't have to start with t = 0 to trace out a full period, though it's common practice to do so. An alternative is to use the roots instead. Or you could go from one peak to an adjacent neighboring peak (or from one valley point to its adjacent neighboring valley). There are many ways to find the period.
Since T = 2pi/3 is the period, this makes,
B = 2pi/T
B = 2pi/(2pi/3)
B = (2pi/1)/(2pi/3)
B = (2pi/1)*(3/2pi)
B = 3
The vertical distance from the midline to either a peak or valley is 3 units, so the amplitude of the graph is 3. This means either A = 3 or A = -3. In this case, A = -3 because the y values on the interval from t = 0 to t = 2pi/6 = pi/3 is negative. Put another way, the y values just after t = 0 are negative.
The phase shift is zero, so C = 0. The midline is also zero, so D = 0.
Using
- A = -3
- B = 3
- C = 0
- D = 0
y = A*sin(B(x-C))+D
y = -3sin(3(x-0))+0
y = -3sin(3x)
so we arrive at the final answer f(t) = -3sin(3t)
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Another way to look at it
Effectively, we start with the parent function y = sin(x) and transform it to y = -3sin(3x) after reflecting it over the x axis, vertically stretching it by a factor of 3, followed finally by a horizontal compression by a factor of 3. This is shown by these graphs below.
y = sin(x) is the parent function (green)
y = -sin(x), shown in red, is the reflection of that parent function over the x axis
y = -3sin(x), shown in blue, is the result of vertically stretching the red function by a factor of 3
y = -3sin(3x), shown in orange, is the result of horizontally compressing the blue curve by a factor of 3
Note: the period of sine is 2pi radians (360 degrees). Going from 2pi radians to 2pi/3 radians means we have divided by 3. This "division by 3" operation is visually represented by a horizontal compression by a factor of 3.