Worked examples
The question
Question: graph y = √x + 4 from y = √x.
y = √x + 4
We know the graph of y = √x.
It is half a U, lying on its side.
we know: y = √x
1 Read the changes
Outside, at the end, + 4 adds 4 to every y.
So the graph moves up 4.
+ 4 outside → up 4
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 up 4.
(0, 0) goes to (0, 4).
(1, 1) goes to (1, 5).
(4, 2) goes to (4, 6).
4 Draw the new graph
Join the new points with the same shape.
This is y = √x + 4.
5 Check
Let's check.
Put x = 0 in the new rule.
We get y = 4: the point (0, 4) we drew.
x = 0: y = √0 + 4 = 0 + 4 = 4
And x = 4 gives y = 6: the point (4, 6).
x = 4: y = √4 + 4 = 2 + 4 = 6
Let's look back.
First: read the changes.
Then: start from y = √x.
Then: move the points.
Then: draw the new graph.
Last: check.
Answer + 4 outside → up 4
▶ Watch it solved
The question
Question: graph y = √(x − 2) + 1 from y = √x.
y = √x − 2 + 1
We know the graph of y = √x.
It is half a U, lying on its side.
we know: y = √x
1 Read the changes
Inside, next to x, we have − 2.
x = 2 makes the inside 0.
So the start point moves to x = 2: right 2.
x − 2 inside → right 2
Outside, at the end, + 1 adds 1 to every y.
So the graph moves up 1.
+ 1 outside → up 1
Careful!
Inside, it goes the opposite way to the sign.
Careful: − 2 inside goes right, not left
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 2.
Then up 1.
(0, 0) goes to (2, 1).
(1, 1) goes to (3, 2).
(4, 2) goes to (6, 3).
4 Draw the new graph
Join the new points with the same shape.
This is y = √(x − 2) + 1.
5 Check
Let's check.
Put x = 2 in the new rule.
We get y = 1: the point (2, 1) we drew.
x = 2: y = √2 − 2 + 1 = √0 + 1 = 0 + 1 = 1
And x = 6 gives y = 3: the point (6, 3).
x = 6: y = √6 − 2 + 1 = √4 + 1 = 2 + 1 = 3
Let's look back.
First: read the changes.
Then: start from y = √x.
Then: move the points.
Then: draw the new graph.
Last: check.
Answer + 1 outside → up 1
▶ Watch it solved
The question
Question: graph y = 2√x from y = √x.
y = 2√x
We know the graph of y = √x.
It is half a U, lying on its side.
we know: y = √x
1 Read the changes
The 2 in front makes every y 2 times as big.
The graph gets steeper.
2 in front → every y × 2
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
(0, 0) stays where it is.
(1, 1) goes to (1, 2).
(4, 2) goes to (4, 4).
4 Draw the new graph
Join the new points with the same shape.
This is y = 2√x.
5 Check
Let's check.
Put x = 0 in the new rule.
We get y = 0: the point (0, 0) we drew.
x = 0: y = 2√0 = 2 × 0 = 0
And x = 4 gives y = 4: the point (4, 4).
x = 4: y = 2√4 = 2 × 2 = 4
Let's look back.
First: read the changes.
Then: start from y = √x.
Then: move the points.
Then: draw the new graph.
Last: check.
Answer 2 in front → every y × 2
▶ Watch it solved
