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{{ printedBook.courseTrack.name }} {{ printedBook.name }} An absolute value function is any function that contains the absolute value of a variable expression. These functions can also be described as any function that is a transformation of the absolute value parent function, $f(x)=∣x∣.$ Since the absolute value of an expression is never negative, the graph of the function $f(x)=∣x∣$ will always lie on or above the $x$-axis. Note that $∣-5∣=5and∣5∣=5.$ This means that the points $(-5,5)$ and $(5,5)$ both lie on $f(x)=∣x∣.$ As it turns out, these points lie directly across from each other. In fact, this symmetry exists for all inverse input values. Thus, absolute value graphs have a distinct V-shape.

Any function belonging to the absolute value function family can be written using the equation $y=a∣x−h∣+k$

where $a,$ $h,$ and $k$ are real numbers and $a =0.$Graph the absolute value function $f(x)=∣x−2∣−3$ using a table of values.

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To graph the given function using a table of values, we can substitute various $x$-values into the rule and solve for the corresponding $y$-values. The absolute value of a number is always the positive value of that number. For instance, $∣-2∣=2.$ Let's first calculate the $y$-value that corresponds with $x=-3.$
We have found that $f(-3)=2.$ Thus, the point $(-3,2)$ lies on the graph. We can find other points on the graph in the same way.

$f(x)=∣x−2∣−3$

Substitute$x=-3$

$f(-3)=∣-3−2∣−3$

SubTermSubtract term

$f(-3)=∣-5∣−3$

AbsNeg$∣-5∣=5$

$f(-3)=5−3$

SubTermSubtract term

$f(-3)=2$

$x$ | $∣x−2∣−3$ | $f(x)$ |
---|---|---|

$-2$ | $∣-2−2∣−3$ | $1$ |

$-1$ | $∣-1−2∣−3$ | $0$ |

$0$ | $∣0−2∣−3$ | $-1$ |

$1$ | $∣1−2∣−3$ | $-2$ |

$2$ | $∣2−2∣−3$ | $-3$ |

$3$ | $∣3−2∣−3$ | $-2$ |

$4$ | $∣4−2∣−3$ | $-1$ |

$5$ | $∣5−2∣−3$ | $0$ |

$6$ | $∣6−2∣−3$ | $1$ |

$7$ | $∣7−2∣−3$ | $2$ |

To draw the graph, we can plot these points, then connect them with a V-shaped curve.

An absolute value equation is an equation that contains the absolute value of a variable expression. An example of this kind of equation is $∣∣∣ 2x_{2}−8∣∣∣ =5.$

As is the case with most equations, these can be solved graphically. This is done by moving all terms except the constant term to one side. The function, which is the non-constant side of the equation, is then graphed and the solution(s) to the equation are found as the point(s) on the graph having the $y$-coordinate that equals the constant.Solve the equation graphically. $∣2x−5∣−3=0$

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When solving an equation graphically, the first step is to rearrange the equation so that the constant term is alone on one side. $∣2x−5∣−3=0⇔∣2x−5∣=3$ The left-hand side of the equation can now be expressed as a function, $f(x)=∣2x−5∣.$ We'll draw the graph of $f.$

The solutions to the equation $y=∣2x−5∣=3$ are the $x$-coordinates of the points on the graph that have the $y$-coordinate $3.$

The equation has the solutions $x=1$ and $x=4.$ We can verify the solutions by testing them in the equation. We'll start with $x=1.$$∣2x−5∣=3$

Substitute$x=1$

$∣2⋅1−5∣=?3$

MultiplyMultiply

$∣2−5∣=?3$

SubTermSubtract term

$∣-3∣=?3$

AbsNeg$∣-3∣=3$

$3=3$

Since $x=4$ makes a true statement, it is also a solution. Thus, the equation has the solutions $x=1andx=4.$

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