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Chapter 1 | Functions and Graphs
canwrite cos x = sin( x + π /2). Similarly, we can view the graph of y = sin x as the graph of y =cos x shifted right π /2 units, and state that sin x =cos( x − π /2). A shifted sine curve arises naturally when graphing the number of hours of daylight in a given location as a function of the day of the year. For example, suppose a city reports that June 21 is the longest day of the year with 15.7 hours and December 21 is the shortest day of the year with 8.3 hours. It can be shown that the function h ( t ) =3.7sin ⎛ ⎝ 2 π 365 ( t −80.5) ⎞ ⎠ +12 is a model for the number of hours of daylight h as a function of day of the year t ( Figure 1.36 ).
Figure 1.36 The hours of daylight as a function of day of the year can be modeled by a shifted sine curve.
Example 1.27 Sketching the Graph of a Transformed Sine Curve
Sketch a graph of f ( x ) =3sin ⎛ ⎝ 2 ⎛
⎞ ⎠
⎞ ⎠ +1.
⎝ x − π 4
Solution This graph is a phase shift of y = sin( x ) to the right by π /4 units, followed by a horizontal compression by a factor of 2, a vertical stretch by a factor of 3, and then a vertical shift by 1 unit. The period of f is π .
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