Trigonometry Transformations of Trigonometric Graphs Transformations of the Cosine Graph
Graph of y=Acos(bx+c) Concepts 1

We will discuss how the constants A, b and c affect the graph of y=A· cos(bx + c).

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The graph of f(x) = A· cos(bx +c), where b is not equal to 0, can be obtained from the graph of the cosine function by performing the following steps: To obtain the graph of f2(x) = cosbx compress/stretch the graph of f1(x) = cosx horizontally to the period . To obtain the graph of f3(x) = cos(bx + c) shift the graph of f2(x) horizontally respect to the phase shift . To obtain the required graph f(x) compress/stretch the graph of f3(x) vertically respect to the amplitude |A|. Reflect the basic cosine curve in the x-axis if A<0.

Example 1 . Graph f(x) = 3· cos(-2x + 4)
Solution
We have A = 3,b = -2, c = 4.
Step 1. To obtain the graph of f2(x) = cos(-2x ) compress the graph of f1(x) = cosx horizontally to the period .
Step 2.
To obtain the graph of f3(x) = cos(-2x +4 ) shift the graph of f2(x) horizontally respect to the phase shift . Shift to the right.
Step 3.
To obtain the required graph f(x) stretch the graph of f3(x) vertically respect to the amplitude |A|=3.

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