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**Unformatted text preview: **gt; 0. 678 Foundations of Trigonometry a shift to the right 1 unit) and the vertical shift is B = 1 (indicating a shift up 1 unit.) All of
these match with our graph of y = f (x). Moreover, if we start with the basic shape of the cosine
graph, shift it 1 unit to the right, 1 unit up, stretch the amplitude to 3 and shrink the period
to 4, we will have reconstructed one period of the graph of y = f (x). In other words, instead of
tracking the ﬁve ‘quarter marks’ through the transformations to plot y = f (x), we can use ﬁve
other pieces of information: the phase shift, vertical shift, amplitude, period and basic shape of the
cosine curve. Turning our attention now to the function g in Example 10.5.1, we ﬁrst need to use
the odd property of the sine function to write it in the form required by Theorem 10.23
g (x) = 1
3
1
3
1
3
1
3
sin(π − 2x) + = sin(−(2x − π )) + = − sin(2x − π ) + = − sin(2x + (−π )) +
2
2
2
2
2
2
2
2 π
We ﬁnd A = − 1 , ω = 2, φ = −π and B = 3 . The period is then 22 = π , the amplitude is
2
2
1
−π
π
π
1
− 2 = 2 , the phase shift is − 2 = 2 (indicating a shift right 2 units) and the vertical shift is up
3
2 . Note that, in this case, all of the data match our graph of y = g (x) with the exception of the
ph...

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