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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Let's start by recalling possible transformations of the parent function $f(x)=x1 .$

Function | Transformation of the Graph of $f(x)=x1 $ |
---|---|

$g(x)=x−h1 $ | Horizontal translation by $h$ units. If $h>0,$ the translation is to the right. If $h<0,$ the translation is to the left. |

$g(x)=x1 +k$ | Vertical translation by $k$ units. If $k>0,$ the translation is up. If $k<0,$ the translation is down. |

$g(x)=a(x1 )$ | Vertical stretch or shrunk by a factor of $a.$ If $a>1,$ it is a vertical stretch. If $0<a<1,$ it is a vertical compression. |

$g(x)=-x1 $ | Reflection across the $x-$axis. |

Now, let's consider the given function. $g(x)=-0.5(x−11 )−2⇕g(x)=-0.5(x−11 )+(-2) $ We can see that $a=-0.5,$ $h=1,$ and $k=-2.$ From here, we can determine the transformations.

- A vertical compression by a factor of $0.5.$
- A reflection across the $x-$axis.
- A horizontal translation $1$ unit to the right.
- A vertical translation $2$ units down.

Using these transformations, we can find the asymptotes and the reference points of the graph of $g(x).$ Note that the horizontal translation affects only the $x-$coordinates, while the vertical compression, reflection across the $x-$axis, and vertical translation affect only the $y-$coordinates.

Object | $f(x)=x1 $ | $g(x)=-0.5(x−11 )+(-2)$ |
---|---|---|

Vertical asymptote | $x=0$ | $x=0+1$ $⇕$ $x=1$ |

Horizontal asymptote | $y=0$ | $y=0+(-2)$ $⇕$ $y=-2$ |

Reference point | $(-1,-1)$ | $(-1+1,-0.5(-1)+(-2))$ $⇕$ $(0,-1.5)$ |

Reference point | $(1,1)$ | $(1+1,-0.5(1)+(-2))$ $⇕$ $(2,-2.5)$ |

Next, we will use the table above to graph $f(x)$ and $g(x).$