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Absolute Value Functions

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from 0 on the number line.

The absolute value parent function, written as f(x) = |x|, is defined as

To graph an absolute value function, choose several values of x and find some ordered pairs.

Plot the points on a coordinate plane and connect them.

Observe that the graph is V-shaped.

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

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

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

(4) The range is the set of al real numbers greater than or equal to 0.  .

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

Vertical Shift

To translate the absolute value function f(x) = |x| vertically, you can use the function

g(x) = f(x) + k .

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

When k < 0, the graph of g(x) translated k units down.

Horizontal Shift

To translate the absolute value function f(x) = |x| horizontally, you can use the function

g(x) = f(x + k).

When k > 0, the graph of g(x) translated k units left.

When k < 0, the graph of g(x) translated k units right.

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Definition

The signum function, denoted , is defined as follows:

Note: In the definition given here, we define the value  to be zero. In some alternative definitions,  is considered to be undefined, i.e., the domain of  is taken to be the nonzero reals.

SPLIT( ) function

Returns a string containing the text between two separators in the specified string.

Syntax

SPLIT(string, separator, segment <,text_qualifier>)

Parameters

string

Character. The value to extract the text string from.

separator

Character. The character or character string that identifies the start of the segment to extract.

segment

Numeric....