Step by step
The question
Question: graph y = (x + 1)² + 4 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
Inside, next to x, we have + 1.
x = −1 makes the inside 0.
So the turning point moves to x = −1: left 1.
Careful!
Inside, it goes the opposite way to the sign.
Outside, at the end, + 4 adds 4 to every y.
So the graph moves up 4.
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 left 1.
Then up 4.
The new turning point is (−1, 4).
4 Draw the new graph
The shape stays the same.
From the turning point, 1 right, 1 up.
And 1 left, 1 up.
Join them with the same U.
This is y = (x + 1)² + 4.
5 Check
Let's check.
Put x = −1 in the new rule.
We get y = 4: the turning point (−1, 4).
And x = 0 gives y = 5: the point (0, 5) 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.