How do you graph #y = 2sin(4x) + 1#, by using the period, domain, range and intercepts?

1 Answer
Dec 7, 2016

a) In the function #y = asin(b(x+ c)) + d#, the period is given by #(2pi)/b#.

Call the period P.

#P = (2pi)/4#

#P = pi/2#

#:.#The period is #pi/2# units.

b) In the function #y = asin(b(x + c)) + d#, the amplitude is #a# and the vertical displacement is #d#.

The maximum is given by #a + d#.

The minimum is given by #a - d#.

Accordingly, the maximum is at #y= 3# and the minimum is at #y = 1#. Therefore, the range is #{y|1 ≤ y ≤ 3, y in RR}#.

c) We have most of the information needed to graph from the above questions, but I recommend you find the intercepts (the y intercept and a general rule for the x-intercepts.

y intercept

#y = 2sin(4 xx 0) + 1#

#y = 2sin(0) + 1#

#y = 0 + 1#

#y = 1#

x intercepts

#0 = 2sin(4x) + 1#

#-1/2 = sin(4x)#

#arcsin(-1/2) = 4x#

#(7pi)/6, (11pi)/6= 4x#

#x = (7pi)/24 and (11pi)/24#

Due to the periodicity of the sine function, #x = (7pi)/24 + 2pin# and #(11pi)/24 + 2pin# where #n# is an integer.

We have all the information necessary to graph. My grapher won't let me label points, but you can look above and find a good many (using x-intercepts, periodicity, maximums and minimums).

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Hopefully this helps!