Question

Suppose the function f(x) = cos(x) is transformed and the new rule for the translated function, g(x), is g(x) = cos[(2/5)x]. Describe this transformation.

Answer 1

The function g(x) = cos[(2/5)x] is a horizontally stretched version of f(x) = cos(x), with the period of the original function being increased by a factor of (5/2). There is no phase shift or change in amplitude involved.

Step-by-step explanation:

The transformation applied to the function f(x) = cos(x) resulting in the new function g(x) = cos[(2/5)x] can be described as a horizontal stretch. When the independent variable x is multiplied by a fraction, in this case (2/5), it results in a horizontal stretching of the graph of the function by a factor reciprocal to that fraction, which is (5/2).

This means that the period of the cosine function will increase; it will take a longer interval on the x-axis for the function to complete one full cycle. The graph of g(x) will look like the original cosine graph, but it will be wider. This type of transformation does not involve a phase shift or a change in amplitude, so the maxima and minima of the function remain at +1 and -1, respectively.