Worked examples
The question
Question: graph y = −√x from y = √x.
y = −√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 minus in front changes the sign of every y.
Up becomes down.
So the graph flips over the x-axis.
− in front → flip over the x-axis
Careful!
A minus inside flips left and right.
A minus outside flips up and down.
Careful: inside → left-right, outside → up-down
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, −1).
(4, 2) goes to (4, −2).
4 Draw the new graph
Join the new points with the same shape, flipped.
This is y = −√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 = −√0 = 0
And x = 4 gives y = −2: the point (4, −2).
x = 4: y = −√4 = −2
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 − in front → flip over the x-axis
▶ Watch it solved
The question
Question: graph y = −x³ + 1 from y = x³.
y = −x3 + 1
We know the graph of y = x³.
It is an S shape.
we know: y = x³
1 Read the changes
The minus in front changes the sign of every y.
Up becomes down.
So the graph flips over the x-axis.
− in front → flip over the x-axis
Outside, at the end, + 1 adds 1 to every y.
So the graph moves up 1.
+ 1 outside → up 1
Careful!
A minus inside flips left and right.
A minus outside flips up and down.
Careful: inside → left-right, outside → up-down
2 Start from y = x³
Draw the grid.
Draw y = x³ first.
It goes through (−1, −1), (0, 0) and (1, 1).
3 Move the points
The middle point (0, 0) moves up 1.
(−1, −1) goes to (−1, 2).
(0, 0) goes to (0, 1).
(1, 1) goes to (1, 0).
4 Draw the new graph
Join the new points with the same shape, flipped.
This is y = −x³ + 1.
5 Check
Let's check.
Put x = 0 in the new rule.
We get y = 1: the point (0, 1) we drew.
x = 0: y = −(0)3 + 1 = 0 + 1 = 1
And x = 1 gives y = 0: the point (1, 0).
x = 1: y = −(1)3 + 1 = −1 + 1 = 0
Let's look back.
First: read the changes.
Then: start from y = x³.
Then: move the points.
Then: draw the new graph.
Last: check.
Answer + 1 outside → up 1
▶ Watch it solved
The question
Question: graph y = −√(x − 1) from y = √x.
y = −√x − 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
The minus in front changes the sign of every y.
Up becomes down.
So the graph flips over the x-axis.
− in front → flip over the x-axis
Inside, next to x, we have − 1.
x = 1 makes the inside 0.
So the start point moves to x = 1: right 1.
x − 1 inside → right 1
Careful!
A minus inside flips left and right.
A minus outside flips up and down.
Careful: inside → left-right, outside → up-down
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.
(0, 0) goes to (1, 0).
(1, 1) goes to (2, −1).
(4, 2) goes to (5, −2).
4 Draw the new graph
Join the new points with the same shape, flipped.
This is y = −√(x − 1).
5 Check
Let's check.
Put x = 1 in the new rule.
We get y = 0: the point (1, 0) we drew.
x = 1: y = −√1 − 1 = −√0 = 0
And x = 5 gives y = −2: the point (5, −2).
x = 5: y = −√5 − 1 = −√4 = −2
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 x − 1 inside → right 1
▶ Watch it solved
