`f(x) = cos(x), g(x) = cos(x + pi)` Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts.

Textbook Question

Chapter 4, 4.5 - Problem 20 - Precalculus (3rd Edition, Ron Larson).
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msubulldog's profile pic

msubulldog | High School Teacher | (Level 1) Adjunct Educator

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Anytime you are given 2 functions f(x) and g(x) where the argument of g is an "adjustment" within the expression in parentheses in the form (x + h, where h is any real number) the graph of the parent function f(x) is shifted h units in the opposite direction to the sign. An example would be if f(x) = x^2 and g(x) = (x-5)^2, then the graph of g is simply the graph of f shifted 5 units to the right, as there is a - sign after x. With trigonometric functions, h represents a phase shift of h units in the opposite direction of the sign in front of it. Therefore g(x) = cos (x + pi) implies a shift of pi units to the left from the graph of parent function f(x) = cos x.

embizze's profile pic

embizze | High School Teacher | (Level 2) Educator Emeritus

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f(x)=cos(x); g(x)=cos(x+pi)

The graph of g(x) is the graph of f(x) translated pi units to the left. This is also described as a phase shift of pi units to the left. The period and amplitude remain unchanged. This is also a reflection in the horizontal axis.

The cosine curve can be described as a sine curve or a sinusoid.

The graph of f(x) in black and g(x) in red:


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