An absolute value function is a role that contains an algebraic expression within absolute worth symbols. Recall the the absolute value of a number is its street from 0 ~ above the number line.

The absolute worth parent function, created as f(x)=|  x  |, is defined as

f(x)={ xifx>00ifx=0−xifx0

To graph one absolute worth function, pick several values of x and find some ordered pairs.

xy=|  x  |
−2 2
−1 1
0 0
1 1
2 2

Plot the point out on a coordinate plane and affix them.

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Observe that the graph is V-shaped.

(1) The crest of the graph is (0,0).

(2) The axis of the contrary (x=0 or y-axis) is the line that divides the graph right into two congruent halves.

(3) The domain is the set of all actual numbers.

(4) The variety is the set of all genuine numbers greater than or equal to 0. That is, y≥0.

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(5) The x-intercept and also the y-intercept space both 0.

Vertical shift

To analyze the pure value function f(x)=|  x  | vertically, you deserve to use the role

g(x)=f(x)+k.

When k>0, the graph of g(x) translated k systems up.

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When k0, the graph of g(x) translated k units down.

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Horizontal transition

To interpret the absolute value duty f(x)=|  x  | horizontally, you deserve to use the function

g(x)=f(x−h).

When h>0, the graph of f(x) is analyzed h systems to the appropriate to obtain g(x).

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When h0, the graph of f(x) is interpreted h units to the left to acquire g(x).

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Stretch and Compression

The stretching or compressing that the absolute value function y=|  x  |is characterized by the duty y=a|  x  |where a is a constant. The graph opens up up if a>0and opens up down as soon as a0.

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For absolute worth equations multiply by a consistent (for example,y=a|  x  |),if 0a1, climate the graph is compressed, and if a>1, it is stretched. Also, if a is negative, then the graph opens downward, rather of upwards as usual.

See more: What Is The Lateral Area Of The Cone To The Nearest Whole Number

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more generally, the type of the equation because that an absolute value function is y=a|  x−h  |+k. Also:

The vertex of the graph is (h,k). The domain of the graph is collection of all real numbers and also the range is y≥kwhen a>0.The domain of the graph is collection of all real numbers and also the range is y≤kwhen a0.The axis of the opposite is x=h.It opens up up if a>0and opens down if a0.The graph y=|  x  |can be interpreted h devices horizontally and k systems vertically to obtain the graph the y=a|  x−h  |+k.The graph y=a|  x  |is broader than the graph of y=|  x  | if |  a  |1and narrow if | a |>1.