Functions with absolute value

The absolute value “flips” the negative part of the graph above the $x$-axis. To draw the graph of a function with a modulus you rewrite it case by case, according to the sign of the argument: in each interval the function becomes an ordinary polynomial, often a parabola.

Example

$y=x^2+2|x|+2x-4$. It is studied case by case: $y=\begin{cases} x^2+4x-4 & \text{per } x\ge0 \\ x^2-4 & \text{per } x<0 \end{cases}$

Example

$y=x|x|-|3+x|+3$. The graph is built piecewise: $y=\begin{cases} -x^2+x+6 & \text{per } x<-3 \\ -x^2-x & \text{per } -3\le x<0 \\ x^2-x & \text{per } x\ge0 \end{cases}$

Links

Topics: Parabola
Concepts: Parabola · Absolute value
Functions: Parabola
Skills: Reasoning by cases · Drawing a graph