Step by step
The question
Question: graph y = 2√(x − 1) + 3 from y = √x.
We know the graph of y = √x.
It is half a U, lying on its side.
1 Read the changes
The 2 in front makes every y 2 times as big.
The graph gets steeper.
Inside, next to x, we have − 1.
x = 1 makes the inside 0.
So the start point moves to x = 1: right 1.
Outside, at the end, + 3 adds 3 to every y.
So the graph moves up 3.
Careful!
Inside, it goes the opposite way to the sign.
2 Start from y = √x
Draw the grid.
Draw y = √x first.
It goes through (0, 0), (1, 1) and (4, 2).
3 Move the points
The start point (0, 0) moves right 1.
Then up 3.
(0, 0) goes to (1, 3).
(1, 1) goes to (2, 5).
(4, 2) goes to (5, 7).
4 Draw the new graph
Join the new points with the same shape.
This is y = 2√(x − 1) + 3.
5 Check
Let's check.
Put x = 1 in the new rule.
We get y = 3: the point (1, 3) we drew.
And x = 5 gives y = 7: the point (5, 7).
Let's look back.
First: read the changes.
Then: start from y = √x.
Then: move the points.
Then: draw the new graph.
Last: check.