Worked examples
The question
Question: graph y = one half times x², from y = x².
y = 12x2
We know the graph of y = x².
It is a U.
we know: y = x²
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.
12 in front → every y × 12
Careful!
A number bigger than 1 makes it narrower.
A number between 0 and 1 makes it wider.
Careful: a number between 0 and 1 in front: wider, not narrower
2 Start from y = x²
Draw the grid.
Draw y = x² first.
It goes through (−2, 4), (0, 0) and (2, 4).
3 Shrink the y's
(−2, 4): one half of 4 is 2.
It goes to (−2, 2).
(0, 0) stays: one half of 0 is 0.
(2, 4): one half of 4 is 2.
It goes to (2, 2).
4 Draw the new graph
Join them with the same shape, wider.
This is the new graph.
5 Check
Let's check: put x = 2 in the new rule.
One half of 4 is 2.
We get the point (2, 2).
It works!
x = 2: y = 12(2)2 = 12 × 4 = 2
Let's look back.
First: read the changes.
Then: start from y = x².
Then: shrink the y's.
Then: draw the new graph.
Last: check.
Answer 12 in front → every y × 12
▶ Watch it solved
The question
Question: graph y = one half times x², + 3, from y = x².
y = 12x2 + 3
We know the graph of y = x².
It is a U.
we know: y = x²
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.
12 in front → every y × 12
Careful!
A number bigger than 1 makes it narrower.
A number between 0 and 1 makes it wider.
Careful: a number between 0 and 1 in front: wider, not narrower
Outside, at the end, + 3 adds 3 to every y.
So the graph moves up 3.
+ 3 outside → up 3
2 Start from y = x²
Draw the grid.
Draw y = x² first.
It goes through (−2, 4), (0, 0) and (2, 4).
3 Shrink the y's
(−2, 4): one half of 4 is 2.
It goes to (−2, 2).
(0, 0) stays: one half of 0 is 0.
(2, 4): one half of 4 is 2.
It goes to (2, 2).
4 Move the turning point
The turning point moves up 3.
Every point moves the same way.
The new points: (−2, 5), (0, 3) and (2, 5).
5 Draw the new graph
Join them with the same shape, wider.
This is the new graph.
6 Check
Let's check: put x = 2 in the new rule.
One half of 4 is 2, plus 3 is 5.
We get the point (2, 5).
It works!
x = 2: y = 12(2)2 + 3 = 12 × 4 + 3 = 2 + 3
= 5
Let's look back.
First: read the changes.
Then: start from y = x².
Then: shrink the y's.
Then: move the turning point.
Then: draw the new graph.
Last: check.
Answer + 3 outside → up 3
▶ Watch it solved
The question
Question: graph y = one half times (x − 2)², − 1, from y = x².
y = 12(x − 2)2 − 1
We know the graph of y = x².
It is a U.
we know: y = x²
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.
12 in front → every y × 12
Careful!
A number bigger than 1 makes it narrower.
A number between 0 and 1 makes it wider.
Careful: a number between 0 and 1 in front: wider, not narrower
Inside, next to x, we have − 2.
x = 2 makes the inside 0.
So the turning point moves right 2.
x − 2 inside → right 2
Outside, at the end, − 1 takes 1 away from every y.
So the graph moves down 1.
− 1 outside → down 1
2 Start from y = x²
Draw the grid.
Draw y = x² first.
It goes through (−2, 4), (0, 0) and (2, 4).
3 Shrink the y's
(−2, 4): one half of 4 is 2.
It goes to (−2, 2).
(0, 0) stays: one half of 0 is 0.
(2, 4): one half of 4 is 2.
It goes to (2, 2).
4 Move the turning point
The turning point moves right 2.
Every point moves the same way.
Then down 1.
Every point moves the same way.
The new points: (0, 1), (2, −1) and (4, 1).
5 Draw the new graph
Join them with the same shape, wider.
This is the new graph.
6 Check
Let's check: put x = 4 in the new rule.
One half of 4 is 2, minus 1 is 1.
We get the point (4, 1).
It works!
x = 4: y = 12(4 − 2)2 − 1 = 12(2)2 − 1
= 12 × 4 − 1 = 2 − 1 = 1
Let's look back.
First: read the changes.
Then: start from y = x².
Then: shrink the y's.
Then: move the turning point.
Then: draw the new graph.
Last: check.
Answer − 1 outside → down 1
▶ Watch it solved
