Sketch a graph of the function. f(x)=4 cos(x+π)

Name: Problem 20, Chapter 0: Calculus: Early Transcendental Functions
Uploaded: 2020-06-02T13:44:19-08:00
Duration: 2 min 40 s
Channel: Amit Srivastava
Description: Sketch a graph of the function. f(x)=4 cos(x+π)

Key Concepts

Cosine Function

The cosine function is a periodic trigonometric function that oscillates between its maximum and minimum values. Its standard graph starts at a maximum point when x = 0 and follows a smooth, continuous wave pattern over a period of 2?. Understanding the basic shape of the cosine curve is essential before applying any transformations.

Amplitude

Amplitude refers to the vertical stretch or compression of the trigonometric graph. It is the absolute value of the coefficient multiplying the cosine function. In this context, the amplitude of 4 means that the peaks of the cosine graph reach 4 and the troughs reach -4, effectively scaling the standard cosine graph vertically.

Period

The period of a trigonometric function is the length of one complete cycle of the graph. For the standard cosine function, the period is 2?. Even though the input is shifted in this function, the period remains unchanged at 2? since there is no horizontal scaling factor modifying the input's coefficient.

Phase Shift

The phase shift is a horizontal translation of the graph. In the function f(x) = 4 cos (x + ?), the expression within the cosine indicates a shift in the x-direction. Specifically, the graph is shifted to the left by ? units. Recognizing phase shifts is crucial for accurately sketching the graph in its new position on the coordinate plane.

Trigonometric Identities

Trigonometric identities, such as cos (x + ?) = -cos x, are key tools for simplifying expressions and understanding transformations. By recognizing that a phase shift of ? in a cosine function results in a reflection about the x-axis, one can verify the graph's behavior and its symmetry properties.

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