Answer:
The transformations needed to obtain the new function are horizontal scaling, vertical scaling and vertical translation. The resultant function is
.
The domain of the function is all real numbers and its range is between -4 and 5.
The graph is enclosed below as attachment.
Step-by-step explanation:
Let be
the base formula, where
is measured in sexagesimal degrees. This expression must be transformed by using the following data:
(Period)
(Minimum)
(Maximum)
The cosine function is a periodic bounded function that lies between -1 and 1, that is, twice the unit amplitude, and periodicity of
radians. In addition, the following considerations must be taken into account for transformations:
1)
must be replaced by
. (Horizontal scaling)
2) The cosine function must be multiplied by a new amplitude (Vertical scaling), which is:
![\Delta z = \frac{z_{max}-z_{min}}{2}](https://tex.z-dn.net/?f=%5CDelta%20z%20%3D%20%5Cfrac%7Bz_%7Bmax%7D-z_%7Bmin%7D%7D%7B2%7D)
![\Delta z = \frac{5+4}{2}](https://tex.z-dn.net/?f=%5CDelta%20z%20%3D%20%5Cfrac%7B5%2B4%7D%7B2%7D)
![\Delta z = \frac{9}{2}](https://tex.z-dn.net/?f=%5CDelta%20z%20%3D%20%5Cfrac%7B9%7D%7B2%7D)
3) Midpoint value must be changed from zero to the midpoint between new minimum and maximum. (Vertical translation)
![z_{m} = \frac{z_{min}+z_{max}}{2}](https://tex.z-dn.net/?f=z_%7Bm%7D%20%3D%20%5Cfrac%7Bz_%7Bmin%7D%2Bz_%7Bmax%7D%7D%7B2%7D)
![z_{m} = \frac{1}{2}](https://tex.z-dn.net/?f=z_%7Bm%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
The new function is:
![z'(x) = z_{m} + \Delta z\cdot \cos \left(\frac{2\pi\cdot x}{T} \right)](https://tex.z-dn.net/?f=z%27%28x%29%20%3D%20z_%7Bm%7D%20%2B%20%5CDelta%20z%5Ccdot%20%5Ccos%20%5Cleft%28%5Cfrac%7B2%5Cpi%5Ccdot%20x%7D%7BT%7D%20%5Cright%29)
Given that
,
and
, the outcome is:
![z'(x) = \frac{1}{2} + \frac{9}{2} \cdot \cos \left(\frac{\pi\cdot x}{90^{\circ}} \right)](https://tex.z-dn.net/?f=z%27%28x%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%20%2B%20%5Cfrac%7B9%7D%7B2%7D%20%5Ccdot%20%5Ccos%20%5Cleft%28%5Cfrac%7B%5Cpi%5Ccdot%20x%7D%7B90%5E%7B%5Ccirc%7D%7D%20%5Cright%29)
The domain of the function is all real numbers and its range is between -4 and 5. The graph is enclosed below as attachment.