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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)=b(x)1 $ | Horizontal stretch or shrunk by a factor of $b.$ If $0<b<1,$ it is a horizontal stretch. If $1<b,$ it is a horizontal shrink. |

Now, let's consider the given function. $g(x)=2(x+2)1 +3⇕g(x)=2(x−(-2))1 +3 $ We can see that $b=21 ,$ $h=-2,$ and $k=3.$ From here, we can determine the transformations.

- A horizontal shrink by a factor of $21 .$
- A horizontal translation $2$ units to the left.
- A vertical translation $3$ units up.

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

Feature | $f(x)=x1 $ | $g(x)=2(x−(-2))1 +3$ |
---|---|---|

Vertical asymptote | $x=0$ | $x=0+(-2)$ $⇕$ $x=-2$ |

Horizontal asymptote | $y=0$ | $y=0+3$ $⇕$ $y=3$ |

Reference point | $(-1,-1)$ | $(21 (-1)+(-2),-1+3)$ $⇕$ $(-2.5,2)$ |

Reference point | $(1,1)$ | $(21 (1)+(-2),1+3)$ $⇕$ $(-1.5,4)$ |

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