Step by step
The question
Question: graph y = minus one half times x², + 3, from y = x².
We know the graph of y = x².
It is a U.
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.
Outside, at the end, + 3 adds 3 to every y.
So the graph moves 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): minus one half of 4 is −2.
It goes to (−2, −2).
(0, 0) stays: one half of 0 is 0.
(2, 4): minus 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, 1), (0, 3) and (2, 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 = 2 in the new rule.
One half of 4 is −2, plus 3 is 1.
We get the point (2, 1).
It works!
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.