A method based on substitution is used to find the 5 key points of the graph of the trigonometric function y = - 0.5 sin ( 4(x+pi/3) ) + 2.5 and sketch over two periods.
The substitution PHI = 4 (x+pi/3) is made and the function is written as
y = - 0.5 sin ( PHI ) + 2.5
PHI is a new variable and takes 5 values in the range [ 0 , 2 PI ]. A table of values is made including the values of y as a function of PHI. The last step is to calculate the x values using the values of PHI.
From the substitution PHI = 4 (x+pi/3) , we can write x = PHI / 4 - pi/3
We, therefore, end with up with a table including the PHI values, the y values, and the x values but of course, the PHI values are not needed in the graph, they were used for calculation purposes.
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