Question

How do you graph `Y= log ( x + 1 ) - 7`?

Answer 1

See explanation.

Explanation:

Start by knowing the graph of `y=log(x)`. It's three key features are its shape, `x`-intercept at `(1,0)`, and vertical asymptote at `x=0`.

Now we take each transformation one at a time and see what happens.

The `x+1` causes the graph to shift 1 unit to the left, changing the location of the asymptote to `x=-1` and changing the `x`-intercept to `(0,0)`.

Now shift this new graph 7 units down because of the `-7`. This doesn't change the asymptote but takes the point that was the `x`-intercept and moves it to `(0,-7)`.

None of these changes alter the shape of the graph (they're rigid transformations). So our new graph has exactly the same shape, a key point at `(0,-7)`, and a vertical asymptote at `x=-1`.