Question

How does changing the function from f(x) = −4 cos 3x to g(x) = −4 cos 3x − 6 affect the range of the function?

Answer 1

Changing the function from `f(x)= -4cos3x` to `g(x)= -4cos3x-6` will affect the range of the function because having a -6 at the end of the function will translate the function 6 units down.

Explanation:

The general form of a cos function can be written like so:

`a cos b (x-c) + d`

a: vertical scale; so if `a > 1`, the function will be vertically stretched. Likewise if `a < 1`, the function will be vertically compressed.

b: horizontal scale; so if `b > 1`, the function will be horizontally compressed. Likewise, if `b < 1`, the function will be horizontally stretched.

c: horizontal shift (left/right); so if c is negative, the function will shift to the left. Likewise if c is positive, the function will shift to the right. Beware, since c is in brackets, so let's say `(x-6)`, the 6 is actually considered to be positive, and so the function will shift right, and not left.

d: vertical shift (up/down); so if d is negative, the function will shift down. Likewise, if d is positive, the function will shift up.

`g(x)= -4cos3x-6` deals with the change of d, and so the function will be translated 6 units down, which will change the range. If the original range of the original function `f(x)= -4cos3x` is `-4 < f(x) < 4`, then the transformed function `g(x)= -4cos3x-6` will have a range of `-10 < g(x) < -2`. This is because since the whole function is translated down, the range will shift along with it.