Step by step
The question
Question: graph y = 1/(x − 3) − 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 − 3.
x = 3 makes the inside 0.
So the center moves to x = 3: right 3.
Outside, at the end, − 1 takes 1 away from every y.
So the graph moves down 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 right 3.
Then down 1.
(−1, −1) goes to (2, −2).
(1, 1) goes to (4, 0).
4 Draw the new graph
1/x never touches the axes.
Those two lines move too.
Now it never touches x = 3 and y = −1.
Join the new points with the same shape.
This is y = 1/(x − 3) − 1.
5 Check
Let's check.
Put x = 4 in the new rule.
We get y = 0: the point (4, 0) we drew.
And x = 2 gives y = −2: the point (2, −2).
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.