Step by step
The question
Question: graph y = 1/(x + 2) + 1 from y = 1/x.
We know the graph of y = 1/x.
It is two curves, one in each corner.
1 Read the changes
Inside, next to x, we have + 2.
x = −2 makes the inside 0.
So the center moves to x = −2: left 2.
Outside, at the end, + 1 adds 1 to every y.
So the graph moves up 1.
Careful!
Inside, it goes the opposite way to the sign.
2 Start from y = 1/x
Draw the grid.
Draw y = 1/x first.
It goes through (−1, −1) and (1, 1).
3 Move the points
The center (0, 0) moves left 2.
Then up 1.
(−1, −1) goes to (−3, 0).
(1, 1) goes to (−1, 2).
4 Draw the new graph
1/x never touches the axes.
Those two lines move too.
Now it never touches x = −2 and y = 1.
Join the new points with the same shape.
This is y = 1/(x + 2) + 1.
5 Check
Let's check.
Put x = −1 in the new rule.
We get y = 2: the point (−1, 2) we drew.
And x = −3 gives y = 0: the point (−3, 0).
Let's look back.
First: read the changes.
Then: start from y = 1/x.
Then: move the points.
Then: draw the new graph.
Last: check.