Vertical Translations: [ Interactive Graph ]
If k is any positive real number then,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD1s98U4FS9yJ57OF1CLOLSOlEj1U38Z44vnKa-RQxIYWxAER9SviUlU3G410hO-oC5hzdSFoJVDzlTKmVvuma71LHX9AcilA1O9PQvIZVn7Orm3AzhxCKd9VHopgXlXwrfEGNnbNvRPQ/s1600/ShiftUp.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihuF4CKr8Yj2jbwSdLacPhB9MdX4u2jQbmJtLhrnr5DGb8GnpDEZg5rgMEZRWTqj9hFovWU-z-yArMqQ2LK0lFwvOByRkQEjs29Goty40jc6RVi37ynBuSPc5OJ8J3pldMomEtCmXPqC8/s1600/ShiftDown.png)
Horizontal Translations: [ Interactive Graph ]
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7gdBmbdfaUwt1HZXjtBWoYn028wxlX4mUaSvIsJc3JZPL-1JdhmvKiDGlZhuU3-PyLtNpTwmIQ0onPrci0zR5gyi6YxfjmLiM3XzAQGQZPObxR5LngDcn_2UcLLzBOOQmuNEE7hBZlRQ/s1600/ShiftLeft.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfbUW71zNUIoiIlSJL3NqH6dm9hCPgHAOi-qoZu6TQbiAzGJECFC8Ia6liff0hwi_FCnfqFC6TTe4nAaYeEwNUdTDBA0DOMBOSYU5EARpae2oNariFqj2DOygnuwp1qNoJ6yPsdQN4x34/s1600/ShiftRight.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn52FOy460rtfpi1Z7QDkxaw_Cdn8S4dzsqhKmdbZ7Qc6Dmik1r42EGirw0-gKFlZZfKEbwUVByplq1N8qNnP2iZVK3ft-osaqB3vH-ChTuZUqD8egq8WNlWI9Y-PHEJJYFuwH2D1nKvs/s1600/ReflectX.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizQ2iyVsMGn8HSDS817Xy9UvnylPYZAevlMDTFk5yEt37gIbY57srf24d_NgHmj_QuQ2KDpaXnOygcEL0llpgsAcId-VQgtTmL53j3tO-AkQ0Z9Hz_GduS0UMKGbXdVCS_3awe8efxIpc/s1600/ReflectY.png)
For the first function f(x) = −sqrt(x) all of the y-values are negative which results in a reflection about the x-axis. For the second function f(x) = sqrt(−x) all of the x-values must be negative thus resulting in a reflection about the y-axis.
Sketch the graph.
The -1 indicates a reflection of the graph of the squaring function f(x) = x^2 about the x-axis. Be sure to graph the squaring function using a dashed curve because it will be used as a guide and is not the answer. Next, reflect all points about the x-axis and draw in the final graph with a solid curve.
General Steps for Graphing Functions using Transformations:
1. Identify and graph the basic function using a dashed curve.
2. Identify any reflections first and sketch them using the basic function as a guide.
3. Identify any translations.
4. Use this information to sketch the final graph using a solid curve.
Graph the function and determine the domain and range.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiMAnAx4qsgIRHVKQ7yzuW8Sf9EXP3TvGVt2ykRUgp7iecFnp4uwI0lSMoiKCsdGNpYhQbpskroiy8ktD6NledPabTmnkaZj8DosqLV0G_TiVsdZYvPLVWri3rOf6i9Weo29iB6n7wwX0/s1600/ex11-2-001a.png)
Example:
y = sqrt(x)
Next, notice the reflection about the y-axis,
y = sqrt(−x)
And finally, we see a shift up 1 unit.
y = sqrt(−x) + 1
Example:
y = x^2
Next, notice the shift right 3 units,
y = (x − 3)^2
And finally, we see a shift down 2 units.
y = (x − 3)^2 − 2
Example:
y = abs(x)
Next, notice the reflection about the x-axis,
y = −abs(x)
Finally we see a shift left 1 unit and down 2 units
y = −abs(x+1) − 2
You Tube Videos:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfc8IL5Aeg6pTGecoYuWDB1BnJ8A4yG6b3hnSiJU71B-ffAT5gWvNwLUzPKFkEu1oCerQ1loZ9qbQwMBQyO2JnJ3rq2W_InUMHu5GMI1izyL9kmrbIoKHwXDRXg614nbTyraejMUByOzE/s1600/vid11-4-3.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjo-p11_JZ0IiNplyZZ1QKUhtgUsLG9hI47xipjo6R9_CFPGxKWpnIAV3um2tnnyIMlWG3zrOafUmBwS2bOFsss-F0bUvtL4dbqt3W0_bQ7TdYz5i8hZt-L0CqkT50vytz3AXPyXvotl4I/s1600/vid11-4-4.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUKRnPt9mZBJdwc9qciBQ_v80SNqyRqEbxKEBFCIDlolKd_P6Fh8dx7rqAYt0P1RaP-i8FZGzFUWoFuJreLLvSw2cOoxVomsePCD5ZgbHGKRSHidCBpA41zsGOOmu-52_n-2pOiwxIjLE/s1600/vid11-4-5.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_h9Ee6DOzuM2bu6ICtGfaQheCKs1IFXdAYO04rmC03cbdsDEyL9O9RJD6Ks2cegoUVThIXfab1SlXPQUA7fnTDoiojW0AFfnJmmF_p0zFrms1inpP4fbSONLw11zFC1zT4iyYlVFfg-Q/s1600/vid11-4-6.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigwGA2VaJrsVGMVq8ZbAmw7zc2iML6WkxHxjqqzlzSGliD7XkJb-CT5lchY2FjDDcNEe77MrDi66pg2q-x7KTFTYSJYAFp5g1Nnf9YOpq1hlUaCir8xld2eup22MB2oaYfOChYNHPUmeY/s1600/vid11-4-7.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_CTuyhMs8hJ9UsQ3sVwG9U_uCZYdI42fQgMPRXzAD19xTlVvTCFwFxLXSuCJLNe7AX-8xphCUcfYevxM324Ic5gL8Jr6Qkz3G_quSIpSjcOaq4XmBFb19M8mXsB2Pp5_G1KKT1abmfBo/s1600/vid11-4-8.png)