Step by step
The question
Question: graph y = |x − 2| − 3 from y = |x|.
We know the graph of y = |x|.
It is a V, with its corner at (0, 0).
1 Read the changes
Inside, next to x, we have − 2.
x = 2 makes the inside 0.
So the corner moves to x = 2: right 2.
Careful!
Inside, it goes the opposite way to the sign.
Outside, at the end, − 3 takes 3 away from every y.
So the graph moves down 3.
2 Start from y = |x|
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Draw y = |x| first.
Its corner is at (0, 0).
3 Move the corner
Move it right 2.
Then down 3.
The new corner is (2, −3).
4 Draw the new graph
The shape stays the same.
From the corner, 1 right, 1 up.
And 1 left, 1 up.
Join them with the same V.
This is y = |x − 2| − 3.
5 Check
Let's check.
Put x = 2 in the new rule.
We get y = −3: the corner (2, −3).
And x = 3 gives y = −2: the point (3, −2) we drew.
Let's look back.
First: read the changes.
Then: start from y = |x|.
Then: move the corner.
Then: draw the new graph.
Last: check.