# How do you graph  f(x) = |x| + |x + 2| ?

Jan 31, 2018

#### Explanation:

Given:

color(blue)(y=f(x) = |x|+|x+2|

We need to graph this absolute value function.

We will assign the values for $\textcolor{red}{x}$ as follows and find the corresponding $\textcolor{red}{y}$ value.

color(blue)(x: +4, +3, +2, +1, " "0," " -1, -2, -3, -4

To find the corresponding $\textcolor{b l u e}{y}$ value, we will substitute (in order) the values of color(red)(x in

color(blue)(y = |x|+|x+2|

Please look at the table of values that contains all values for color(red)(x and y: and also as Ordered Pair to facilitate graphing.

The corresponding graph is given below:

Observations:

For the parent function $f \left(x\right) = | x |$,

Vertex is at $\left(0 , 0\right)$

Vertex is the minimum point on the graph.

Axis of Symmetry is at color(green)(x = 0

A translation is a transformation that shifts a graph horizontally or vertically, but does not change the orientation of the graph.

Please refer to the graph to observe the following:

Domain of the function

color(blue)(y=f(x) = |x|+|x+2|

is color(green)((-oo, oo)

Range of the function

color(blue)(y=f(x) = |x|+|x+2|

is color(brown)([2, oo), since color(green)(f(x) >= 2

Extreme Points are none for $f \left(x\right)$

Critical Points:

Critical points are points where the function is defined and its derivative is zero or undefined.

Critical Points are at color(green)(x=-2, x=0

X and Y Intercepts:

x-axis interception points of color(blue)(f(x) = |x|+|x+2|: None

y-axis interception points of color(blue)(f(x) = |x|+|x+2|: (0, 2)

The graph of color(blue)(y=f(x) = |x|+|x+2|

has shifted up (a.k.a. Vertical Translation) by two units comparing to the graph of the parent function

color(blue)(y=f(x) = |x|