Consider the trigonometric function f(t) =
-3+4cos(pi/3(t-3/2))
What is the amplitude/period of f(t)?
What are the maximum and minimum values attained by f(t)?
the amplitude is 3
you can determine the max/min by simply observing the properties of the cosine curve
the max of cos(anything) is 1 and its minimum is -1
so max of 4cos(.....) is s4 and its minimum is -4
so max of -3 + 4cos(....) is 1 and its min is -7
To find the amplitude and period of the trigonometric function f(t) = -3 + 4cos(π/3(t-3/2)), we need to analyze its equation.
The general form of a cosine function is f(t) = A*cos(B(t - C)) + D, where A represents the amplitude, B represents the coefficient of t that affects the period, C represents the phase shift, and D represents the vertical shift.
In our case, f(t) = -3 + 4cos(π/3(t-3/2)). Comparing this to the general form, we can determine the values of A, B, C, and D.
- Amplitude (A): The amplitude represents the maximum displacement from the mean. In this case, A = 4, so the amplitude of the function is 4.
- Period (P): The period of a cosine function is determined by the coefficient of t inside the parentheses. In this case, B = π/3, which means the period (P) is given by the formula P = 2π/B. Therefore, P = 2π / (π/3) = 2(3) = 6.
- Maximum and Minimum Values: The maximum and minimum values of a cosine function can be found by adding or subtracting the amplitude from the vertical shift (D). In this case, since the vertical shift (D) is -3 and the amplitude (A) is 4, the maximum value is -3 + 4 = 1, and the minimum value is -3 - 4 = -7.
To summarize:
- The amplitude of f(t) is 4.
- The period of f(t) is 6.
- The maximum value attained by f(t) is 1.
- The minimum value attained by f(t) is -7.