4.2 Graphs of Absolute Valued Functions

4.2 Graphs of Absolute Valued Functions

An absolute valued function is a type of function that includes an algebraic expression enclosed within absolute value bars. Its general form is $f(x) = |x|$. The absolute value of a number represents its distance from zero on the number line and is always non-negative. The graph of $f(x) = |x|$ has V-shape and is symmetric with respected to the $y$-axis. The point where the graph changes direction is called the vertex $(0, 0)$. The vertex form of an absolute value function is $g(x) = a|x - h| + k$, where $a \neq 0$ and $(h, k)$ is the vertex. Its graph is symmetric about the line $x = h$.

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Remember!
The graphs of all other absolute value functions are variations of the parent function $f(x) = |x|$ obtained through transformations.

$|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$ [Graph of absolute value function]
Figure 4.12

Example 10

Plot the graph of the following functions: (i) $f(x) = |x| + 2$ (ii) $g(x) = |x - 3|$ Solution (i) $f(x) = |x| + 2$ (ii) $g(x) = |x - 3|$ Step I Make a table of values.

Step II Plot the ordered pairs. Step III Join the ordered pairs and draw the V-shaped graph.

Step I Make a table of values.

Step II Plot the ordered pairs. Step III Join the ordered pairs and draw the V-shaped graph.

[Graph of f(x) = |x| + 2]
Figure 4.13 [Graph of g(x) = |x - 3|]
Figure 4.14

Example 11

Plot the graph of $f(x) = 2|x + 1| - 3$. Solution $f(x) = 2|x + 1| - 3$ Step I Make a table of values.

Step II Plot the ordered pairs. Step III Join the ordered pairs and draw the V-shaped graph of the function. Notice that the vertex of the graph of $f$ is $(-1, -3)$ and the graph is symmetric about $x = -1$. [Graph of f(x) = 2|x+1| - 3]
Figure 4.15