Absolute Value Function: Graph

How to graph an absolute value function on a coordinate plane: 5 examples and their solutions.

Example 1: y = |x|

Example

Solution

See y = |x|.
There's |x|.

Absolute Value

So draw y = x
at x > 0.
(the right side of x = 0).

There's |x|.

So draw the image of the graph
under the reflection in the y-axis.

This is the graph of y = |x|.

Example 2: y = f(|x|)

Example

Solution

See y = 2|x| - 3.
There's |x|.

So draw y = 2x - 3
at x > 0.
(the right side of x = 0).

Slope-Intercept Form

There's |x|.

So draw the image of the graph
under the reflection in the y-axis.

This is the graph of y = 2|x| - 3.

Example 3: |y| = f(x)

Example

Solution

See |y| = 2x - 3.
There's |y|.

So draw y = 2x - 3
at y > 0.
(the upper side of y = 0).

There's |y|.

So draw the image of the graph
under the reflection in the x-axis.

This is the graph of |y| = 2x - 3.

Example 4: |y| = f(|x|)

Example

Solution

See |y| = 2|x| - 3.
There are |x| and |y|.

So draw y = 2x - 3
at the quadrant I.
(x > 0 and y > 0).

There are |x| and |y|.

So draw the images of the graph
under the reflection
in the x-axis,

in the y-axis,

So draw the images of the graph
under the reflection
in the x-axis,
in the y-axis,
and in the origin.

This is the graph of |y| = 2|x| - 3.

Example 5: y = |f(x)|

Example

Solution

See y = |x2 - 4|.
This is y = |f(x)|.

Then draw, y = f(x),
y = x2 - 4.

At the region below the x-axis,
lightly draw the graph.

Quadratic Function: Vertex Form

There's |f(x)|.

Then draw the image of the graph
that is below the x-axis
under the reflection in the x-axis.

Remove the graph below the x-axis.

Then this is the graph of y = |x2 - 4|.