Step by step
The question
Question: graph y = minus one half times √(x − 1), + 2, from y = √x.
We know the graph of y = √x.
It is half a U, lying on its side.
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.
The minus in front changes the sign of every y.
So the graph flips upside down.
Inside, next to x, we have − 1.
x = 1 makes the inside 0.
So the start point moves right 1.
Outside, at the end, + 2 adds 2 to every y.
So the graph moves up 2.
2 Start from y = √x
Draw the grid.
Draw y = √x first.
It goes through (0, 0) and (4, 2).
3 Shrink the y's
(0, 0) stays: one half of 0 is 0.
(4, 2): minus one half of 2 is −1.
It goes to (4, −1).
4 Move the start point
The start point moves right 1.
Every point moves the same way.
Then up 2.
Every point moves the same way.
The new points: (1, 2) and (5, 1).
5 Draw the new graph
Join them with the same shape, wider and upside down.
This is the new graph.
6 Check
Let's check: put x = 5 in the new rule.
One half of 2 is −1, plus 2 is 1.
We get the point (5, 1).
It works!
Let's look back.
First: read the changes.
Then: start from y = √x.
Then: shrink the y's.
Then: move the start point.
Then: draw the new graph.
Last: check.