# How do you graph f(x)=2|x-1|-3?

Oct 3, 2016

Use the graph of the parent function $f \left(x\right) = \left\mid x \right\mid$ but change the slope to of each half of the graph to $\pm 2$, and shift the vertex right one unit and down three units.

#### Explanation:

The parent function of absolute value is $f \left(x\right) = \left\mid x \right\mid$ and looks like
the graph below. Note the vertex is at $\left(0 , 0\right)$ and the slope of the left branch of the graph is $- 1$ while the slope of the right branch is $+ 1$.
graph{abs(x) [-10, 10, -5, 5]}

Given the form, $f \left(x\right) = m \left\mid x - h \right\mid + k$,

$\pm m$ represents the slope of each branch of the graph. A positive value of $m$ in the original equation means the graph has a "V" shape, while a negative value of $m$ means the graph has an "upside-down V-shape".

$\left(h , k\right)$ is the vertex. The vertex is shifted horizontally by h units and vertically by k units.

$f \left(x\right) = 2 \left\mid x - 1 \right\mid - 3$

In this example, $m = 2$ is positive, and the graph will have a V-shape. The left branch of the V has a slope of $- 2$ and the right branch has a slope of $2$.

The vertex is $\left(1 , - 3\right)$, which represents a shift of one unit to the right and 3 units down.
graph{2abs(x-1)-3 [-10, 10, -5, 5]}