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

Dec 20, 2017

See explanation.

#### Explanation:

Start by knowing the graph of $y = \log \left(x\right)$. It's three key features are its shape, $x$-intercept at $\left(1 , 0\right)$, 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 $\left(0 , 0\right)$.

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 $\left(0 , - 7\right)$.

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 $\left(0 , - 7\right)$, and a vertical asymptote at $x = - 1$.