Question

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

Answer 1

Please read the explanation.

Explanation:

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|`