Step by step
The question
Question: graph y = −x² + 3 from y = x².
We know the graph of y = x².
It is a U, with its turning point at (0, 0).
1 Read the changes
The minus in front turns every y into minus y.
So the U 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.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Draw y = x² first.
Its turning point is at (0, 0).
3 Move the turning point
Move it up 3.
The new turning point is (0, 3).
4 Draw the new graph
The shape is flipped.
From the turning point, 1 right, 1 down.
And 1 left, 1 down.
Join them with an upside-down U.
This is y = −x² + 3.
5 Check
Let's check.
Put x = 0 in the new rule.
We get y = 3: the turning point (0, 3).
And x = 1 gives y = 2: the point (1, 2) we drew.
Let's look back.
First: read the changes.
Then: start from y = x².
Then: move the turning point.
Then: draw the new graph.
Last: check.