# How do you graph y=log_2(x-1)+3?

Aug 3, 2016

Step 1: Determine the coordinates of any asymptotes, the domain and the range

Let's start with what's easiest. The range of every logarithmic function is $y \in \mathbb{R}$, that's to say the function is defined to all the real numbers in the y axis.

The domain is a little trickier. It is now that asymptotes come into play. A vertical asymptote will occur when ${\log}_{2} \left(0\right) = \log \frac{0}{\log} 2$ appears. Just like in rational functions, we can determine the equations of any asymptotes by setting the function to 0. However, here, it is only the $x - 1$ we're interested in, since it is contained by the $\log$ in base $2$.

$x - 1 = 0$

$x = 1$

Hence, there will be a vertical asymptote at $x = 1$.

Step 2: Determine any reflections, compressions, stretches and horizontal and/or vertical tranformations

This is a part that is important, but that many people choose to neglect. They will instead use a table of values, which can be helpful. But let's first look at the transformations that appear when converting the function $y = {\log}_{2} \left(x\right)$ into $y = {\log}_{2} \left(x - 1\right) + 3$.

First, you'll notice that the graph has undergone a horizontal transformation of $1$ unit to the right. Then you'll notice it has undergone a vertical transformation of 3 units up (this is if you have taken the chapter on transformations of functions, of course). If you haven't, it would be helpful to learn how to transform functions.

Now that we know this, we can plot a table of values and understand the trend relative to ${\log}_{2} \left(x\right)$.

After doing this, you'll end up with the following graph:

Hopefully this helps!