Step by step
The question
Question: graph y = −log₂(x + 2) + 1.
A log is a power.
log₂ of c is the power of 2 that makes c.
1 Find the domain
2 to any power is always more than 0.
So a log only takes positive numbers.
The inside must be more than 0.
The inside is x + 2.
So x + 2 > 0.
To remove the + 2, we take away 2 on both sides.
So the domain is x > −2.
Books also write it as (−2, ∞).
The round bracket means −2 itself is not in.
∞ means it goes on for ever.
2 Vertical asymptote
At the edge, the inside is 0: x = −2.
Close to it, the graph goes up and up.
This is the vertical asymptote.
Careful!
The asymptote is where the inside is 0, not where x is 0.
Here that is x = −2.
3 Find some points
The log is easy when the inside is a power of 2: 1/2, 1, 2 and 4.
We choose x to make that happen.
Make the inside 1/2: x is −3/2.
log₂ of 1/2 is −1, because 2 to the power −1 is 1/2.
So y is 2.
Inside 1: x is −1.
log₂ of 1 is 0, because 2 to the power 0 is 1.
So y is 1.
Inside 2: x is 0.
log₂ of 2 is 1, because 2 to the power 1 is 2.
So y is 0.
Inside 4: x is 2.
log₂ of 4 is 2, because 2 to the power 2 is 4.
So y is −1.
4 Draw the graph
Draw the grid.
The asymptote x = −2, dashed.
Plot the points.
Join them with a smooth curve.
Near x = −2 it goes up along the dashed line, but never touches it.
So: domain x > −2, asymptote x = −2.
5 Check
Let's check: at x = −2 the inside is 0, and no power of 2 is 0.
So there is no point there: the asymptote is right.
The purple curve is y = log₂ x, the plain log.
Ours is that curve flipped upside down by the minus, and moved left 2 and up 1.
The same shape.
It works!
Let's look back.
First: find the domain.
Then: vertical asymptote.
Then: find some points.
Then: draw the graph.
Last: check.