An pure value role is a role that contains an algebraic expression within absolute worth symbols. Recall that the absolute worth of a number is its distance from 0 top top 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, select several values of x and also find part ordered pairs.

 x y=|  x  | −2 2 −1 1 0 0 1 1 2 2

Plot the points on a coordinate plane and connect them.

Observe the the graph is V-shaped.

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

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

(3) The domain is the collection of all genuine numbers.

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

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(5) The x-intercept and 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 the g(x) analyzed k systems up.

When k0, the graph the g(x) analyzed k systems down.

### Horizontal change

To analyze the pure value role f(x)=|  x  | horizontally, you can use the role

g(x)=f(x−h).

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

When h0, the graph that f(x) is interpreted h systems to the left to get g(x).

## Stretch and Compression

The extending or compressing the the pure 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 once a0.

For absolute value equations multiply by a continuous (for example,y=a|  x  |),if 0a1, climate the graph is compressed, and also if a>1, the is stretched. Also, if a is negative, climate the graph opens up downward, instead of upwards as usual.

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more generally, the form of the equation for an pure value function is y=a|  x−h  |+k. Also:

The vertex of the graph is (h,k). The domain that the graph is collection of all actual numbers and also the selection is y≥kwhen a>0.The domain the the graph is set of all actual numbers and the variety is y≤kwhen a0.The axis of the contrary is x=h.It opens up if a>0and opens down if a0.The graph y=|  x  |can be interpreted h units horizontally and k systems vertically to gain the graph of y=a|  x−h  |+k.The graph y=a|  x  |is more comprehensive than the graph of y=|  x  | if |  a  |1and narrower if | a |>1.