Reflecting the graph of y = cos x across the y-axis is the same as not reflecting it at all.
A. True
B. False
Answer (2)
Recognize that reflecting a function across the y-axis means replacing x with − x .
Recall that the cosine function is an even function, meaning cos ( x ) = cos ( − x ) .
Conclude that reflecting y = cos x across the y-axis results in the same graph.
Therefore, the statement is True .
Explanation
Problem Analysis The question asks whether reflecting the graph of y = cos x across the y-axis changes the graph.
Reflection Across the y-axis Recall that reflecting a function across the y-axis means replacing x with − x . So, we need to determine if cos ( x ) = cos ( − x ) .
Even Function Property The cosine function is an even function, which means that cos ( x ) = cos ( − x ) for all x . This is a fundamental property of the cosine function.
Conclusion Since cos ( x ) = cos ( − x ) , reflecting the graph of y = cos x across the y-axis results in the same graph. Therefore, reflecting the graph of y = cos x across the y-axis is the same as not reflecting it at all.
Examples
In signal processing, cosine functions are used to model various signals. The property that cos(x) = cos(-x) implies that the signal is symmetric in time. This symmetry simplifies analysis and processing of such signals, making it easier to extract relevant information or manipulate the signal for specific applications.
Reflecting the graph of y = cos x across the y-axis does not change the graph, as cos ( x ) = cos ( − x ) . This means the statement is true. Thus, the answer is True .
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