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

Given:

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

We need to graph this absolute value function.

We will assign the values for #color(red)x# as follows and find the corresponding #color(red)y# value.

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

To find the corresponding #color(blue)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(x) = |x|#,

Vertex is at #(0,0)#

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(x)#

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|#