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{{ printedBook.courseTrack.name }} {{ printedBook.name }} We want to write an equation for the absolute value function represented in the given graph. To do so, we need to determine what *translations* of the parent function $f(x)=∣x∣$ took place. We can use the *vertex form* of absolute value function to create the foundation of our desired equation.
$g(x)=a∣x−h∣+k $
In this form, the constants represent one of the three basic types of transformations.

Variable | Value of the Variable | Transformation |
---|---|---|

$a$ | $a<-1$ | stretch $+$ reflection |

$-1<a<0$ | compression $+$ reflection | |

$0<a<1$ | compression | |

$1<a$ | stretch | |

$h$ | $h<0$ | translation to the left |

$h>0$ | translation to the right | |

$k$ | $k<0$ | translation down |

$k>0$ | translation up |

Looking at the given graph, we can notice that it has not been stretched nor compressed. When there is no stretch nor compression, we have that $a=1.$ This also means that we only need to consider vertical and horizontal translations. Let's compare the given graph with the graph of $f(x)=∣x∣.$

The graph of the parent function has been translated left $2$ units and down $1$ unit. We can substitute these values, as well as $a=1,$ into the general vertex form to find the equation of the function. $g(x)=1∣x+2∣−1⇒g(x)=∣x+2∣−1 $