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Absolute Value Function

Summary — Absolute Value Function f(x) = |x|

The graph of f(x) = |x| has a shape.

• A negative in front of the absolute value, as in f(x) = -|x|, causes the vee to be inverted, like this .

• A number added or subtracted inside the absolute value symbols, as in f(x) = |x + h| or f(x) = |x - h|, shifts the horizontally by h units.

For example:

• The graph of f(x) = |x + 4| is shifted 4 units to the left.

• The graph of f(x) = |x - 3| is shifted 3 units to the right.

• A number added or subtracted outside the absolute value symbols, as in f(x) = |x| + k or f(x) = |x| - k, shifts the vertically by k units.

For example:

• The graph of f(x) = |x| + 2 is shifted 2 units up.

• The graph of f(x) = |x| - 1 is shifted 1 unit down.

To determine the direction of a horizontal shift, it may be helpful to rewrite the expression inside the absolute value symbols in the following ways:

 

Form: f(x)

Example: f(x)

 

= |x + h|

= | x + 4|

= | x - (-4)|

Form: f(x)

Example: f(x)

= |x - h|

= |x - 3|

= |x - (+3)|

Here, h = -4. So, the graph is shifted 4 units to the left. Here, h = +3. So, the graph is shifted 3 units to the right.
 

Example

Graph f(x) = |x + 2| + 3. Determine its domain and range.

Solution

We could calculate and plot points. However, from the above discussion we know that this graph will be the same as the graph of f(x) = |x| but shifted 2 units to the left (the 2 is added to the x) and 3 units up (the 3 is added to the absolute value).

The domain is all real numbers.

From the graph we can see that the y-coordinate of the vertex is 3 and the vee opens upward. Therefore, range is y 3.

 
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