College Algebra & Trigonometry, 2018a

420 CHAPTER 9. GRAPHING THE TRIGONOMETRIC FUNCTIONS
shifted the graph a distance of π to the left and made the new starting point of
the sine curve −π.
In graphing the standard sine curve we’re generally interested in the quadrantal
angles that produce the maximum, minimum and zero points of the graph. In
graphing the function y = sin(x+ π 3
), we want to know which values of x will
produce the quadrantal angles when we add π 3
to them.
So, to determine the new starting point we want to know the solution to the
equation: x+ π 3 =0 x + π 3 =0
− π 3 − π 3
x =− π 3
This is the new starting point for the graph y = sin(x+ π 3
). Because this graph
has a standard period, the “jump” between each of the quadrantal angles will be
π
2
. To graph one period of a typical trigonometric function we’ll need at least five
quadrantal angle values. So, if our new starting point is − π 3
, then the next critical
value along the x-axis will be:
− π 3 + π 2 = −2π 6 + 3π 6 = π 6
Then the subsequent critical values would be:
π
6 + π 2 = π 6 + 3π 6 = 4π 6 = 2π 3
4π
6 + 3π 6 = 7π 6
7π
6 + 3π 6 = 10π
6
= 5π 3
So the five critical values along the x-axis are: