What is the range (maximum and minimum values) of: y = 2 cos (4x + 1)

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

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Find the range of `y=2cos(4x+1)`

We can rewrite as `y=2cos(4(x+1/4))`

This is in the form `y=acos(b(x-h))+k`

a affects the amplitude (and also reflects across the x-axis if negative)

b affects the period

h is a horizontal translation (phase shift)

k is a vertical translation

For this problem a=2 so the base function y=cos(x) is stretched vertically by a factor of 2.

b=4 so the period is `pi/2`

h=1/4 so there is a translation of teh base function 1/4 units left


Since k=0, the midline is y=0. The maximum of the base function is 1; since a=2 the maximum of the transformed equation is 2. The minimum of the base function is -1; the minimum of the transformed equation is -2.


The range of the function `y=2cos(4x+1)` is R=4. The maximum is 2, and the minimum is -2.


The graph: