The graph of [tex]y=4 sin (x+3)-2[/tex] is obtained by shifting the graph of [tex]y=4 sin x-2[/tex] horizontally 3 units to the right.
A. True
B. False
Answer (2)
The problem involves analyzing the horizontal shift of a sine function.
The given functions are y = 4" , " s in x − 2 and y = 4" , " s in ( x + 3 ) − 2 .
The transformation x → x + 3 represents a horizontal shift of 3 units to the left.
Therefore, the statement is false: F a l se .
Explanation
Understanding the Problem We are given two functions: y = 4" , " s in x − 2 and y = 4" , " s in ( x + 3 ) − 2 . We need to determine if the second function is a horizontal shift of the first function by 3 units to the right.
General Form of Sine Function The general form of a sine function is y = A " , " s in ( B x − C ) + D , where:
A is the amplitude
B affects the period
C is the horizontal shift (phase shift)
D is the vertical shift
Comparing the Functions In our case, we have y = 4" , " s in x − 2 and y = 4" , " s in ( x + 3 ) − 2 . Comparing the two functions, we can see that the amplitude and vertical shift are the same. The only difference is the argument of the sine function.
Determining the Shift For the first function, the argument is x . For the second function, the argument is x + 3 . The transformation $x ",
Conclusion Therefore, the graph of y = 4" , " s in ( x + 3 ) − 2 is obtained by shifting the graph of y = 4" , " s in x − 2 horizontally 3 units to the left , not to the right.
Examples
Understanding horizontal shifts of trigonometric functions is crucial in fields like signal processing and physics. For example, in analyzing sound waves, a horizontal shift represents a time delay. If you have a sound wave represented by y = A " , " s in ( t ) , a time delay of 2 seconds would be represented by y = A " , " s in ( t − 2 ) , shifting the graph to the right. Similarly, in electrical engineering, understanding phase shifts is essential for designing circuits and analyzing AC signals.
The statement claiming that y = 4 sin ( x + 3 ) − 2 represents a 3-unit right shift from y = 4 sin x − 2 is false. Instead, it is shifted 3 units to the left. Therefore, the correct answer is B. False.
;
