Consider the function f(x)=2sin⁡(π2(x−3))+4. State the amplitude A, period P, and midline. State the phase shift and vertical translation. In the full period [0, P], state the maximum and minimum y-values and their corresponding x-values. Hints for the maximum and minimum values of f(x): The maximum value of y=sin⁡(x) is y=1 and the corresponding x values are x=π2 and multiples of 2π less than and more than this x value. You may want to solve π2(x−3)=π2. The minimum value of y=sin⁡(x) is y=−1 and the corresponding x values are x=3π2 and multiples of 2π less than and more than this x value. You may want to solve π2(x−3)=3π2. If you get a value for x that is less than 0, you could add multiples of P to get into the next cycles. If you get a value for x that is more than P, you could subtract multiples of P to get into the previous cycles. For x in the interval [0, P], the maximum y-value and corresponding x-value is at: x= y= For x in the interval [0, P], the minimum y-value and corresponding x-value is at: x= y=

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Consider the function f(x)=2sin⁡(π2(x−3))+4.  State the amplitude A, period P, and midline. State the phase shift and vertical translation. In the full period [0, P], state the maximum and minimum y-values and their corresponding x-values.   Hints for the maximum and minimum values of f(x):   The maximum value of y=sin⁡(x) is y=1 and the corresponding x values are x=π2 and multiples of 2π less than and more than this x value. You may want to solve π2(x−3)=π2.   The minimum value of y=sin⁡(x) is y=−1 and the corresponding x values are  x=3π2 and multiples of 2π less than and more than this x value. You may want to solve π2(x−3)=3π2.   If you get a value for x that is less than 0, you could add multiples of P to get into the next cycles.   If you get a value for x that is more than P, you could subtract multiples of P to get into the previous cycles.    For x in the interval [0, P], the maximum y-value and corresponding x-value is at:   x=   y=   For x in the interval [0, P], the minimum y-value and corresponding x-value is at:   x=   y=

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Step 1 Consider a given function  f(x)=2sin⁡[π2(x−3)+4].  State the amplitude A, period P and mid-line.  State the phase shift and vertical translation.  State the minimum and maximum y-values and their corrosponding x-values in the interval [0,P].  Step 2  a) Consider a given function  f(x)=2sin⁡[π2(x−3)]+4  The amplitude of the function f is 2. Hence, A=2.  The period of the function f is 2π(π2)=4. Hence, T=4.  The mid-line of the trigonometric function is a point where the function attain its maximum values.  Now,  π2(x−3)=(2n−1)π2  x−3=2n−1  ⇒x=2n+2 for all n∈Z  Hence, the mid-line of the function f is x=2n+2 for all n∈Z  Step 3  b) Consider a given function  f(x)=2sin⁡[π2(x−3)]+4  The function f is shifted right side by 3 units. Hence, its phase shift is 3.  The function f is shifted vertically upward by 4 units. Hence, its vertically shift is 4 units.  Step 4  c)  The range of the trigonometric function is [−1,1].  It implies that −1≤sin⁡(x)≤1.  Hence,  −1≤sin⁡(π2(x−3))≤1  −2≤2sin⁡(π2(x−3))≤2  −2+4≤2sin⁡(π2(x−3))+4≤2+4  2≤f(x)≤6  Hence, minimum value of the function f is 2 and the maximum value of the function f is 6.  It implies that fmin=2 and fmax=6.  Step 5  Further,  Maximum attain at x=2n+2 for all n∈Z.  Hence, the point of maxima is x=0,2,4.  sin⁡(π2(x−3))=0  π2(x−3)=nπ  x−3=2n  ⇒x=2n+3 for all n∈Z  Hence, the point of minima is 1,3.

Consider the function f(x)=2 \sin(\pi2(x−3))+4. State the amplitude A

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