Step by step
The question
Question: graph y = one half times |x + 3|, − 1, from y = |x|.
We know the graph of y = |x|.
It is a V.
1 Read the changes
The one half in front makes every y one half as big.
Smaller y's: the graph gets wider and flatter.
Careful!
A number bigger than 1 makes it narrower.
A number between 0 and 1 makes it wider.
Inside, next to x, we have + 3.
x = −3 makes the inside 0.
So the corner moves left 3.
Outside, at the end, − 1 takes 1 away from every y.
So the graph moves down 1.
2 Start from y = |x|
Draw the grid.
Draw y = |x| first.
It goes through (−2, 2), (0, 0) and (2, 2).
3 Shrink the y's
(−2, 2): one half of 2 is 1.
It goes to (−2, 1).
(0, 0) stays: one half of 0 is 0.
(2, 2): one half of 2 is 1.
It goes to (2, 1).
4 Move the corner
The corner moves left 3.
Every point moves the same way.
Then down 1.
Every point moves the same way.
The new points: (−5, 0), (−3, −1) and (−1, 0).
5 Draw the new graph
Join them with the same shape, wider.
This is the new graph.
6 Check
Let's check: put x = −1 in the new rule.
One half of 2 is 1, minus 1 is 0.
We get the point (−1, 0).
It works!
Let's look back.
First: read the changes.
Then: start from y = |x|.
Then: shrink the y's.
Then: move the corner.
Then: draw the new graph.
Last: check.