Question

Base: z(x)=cosx Period:180 Maximum:5 Minimum: -4 What are the transformation? Domain and Range? Graph?

Answer 1

The transformations needed to obtain the new function are horizontal scaling, vertical scaling and vertical translation. The resultant function is
`z'(x) = (1)/(2) + (9)/(2) \cdot \cos \left((\pi\cdot x)/(90^(\circ)) \right)`.

The domain of the function is all real numbers and its range is between -4 and 5.

The graph is enclosed below as attachment.

Explanation:

Let be
`z (x) = \cos x` the base formula, where
`x` is measured in sexagesimal degrees. This expression must be transformed by using the following data:

`T = 180^(\circ)` (Period)

`z_(min) = -4` (Minimum)

`z_(max) = 5` (Maximum)

The cosine function is a periodic bounded function that lies between -1 and 1, that is, twice the unit amplitude, and periodicity of
`2\pi` radians. In addition, the following considerations must be taken into account for transformations:

1)
`x` must be replaced by
`(2\pi\cdot x)/(180^(\circ))`. (Horizontal scaling)

2) The cosine function must be multiplied by a new amplitude (Vertical scaling), which is:

`\Delta z = (z_(max)-z_(min))/(2)`

`\Delta z = (5+4)/(2)`

`\Delta z = (9)/(2)`

3) Midpoint value must be changed from zero to the midpoint between new minimum and maximum. (Vertical translation)

`z_(m) = (z_(min)+z_(max))/(2)`

`z_(m) = (1)/(2)`

The new function is:

`z'(x) = z_(m) + \Delta z\cdot \cos \left((2\pi\cdot x)/(T) \right)`

Given that
`z_(m) = (1)/(2)`,
`\Delta z = (9)/(2)` and
`T = 180^(\circ)`, the outcome is:

`z'(x) = (1)/(2) + (9)/(2) \cdot \cos \left((\pi\cdot x)/(90^(\circ)) \right)`

The domain of the function is all real numbers and its range is between -4 and 5. The graph is enclosed below as attachment.