# 005 Sample Final A, Question 17

Question Graph the following function,

${\displaystyle f(x)=\log _{2}(x+1)+2}$

Make sure to label any asymptotes, and at least two points on the graph.

Foundations
1) What is the basic graph of ${\displaystyle f(x)=\log _{2}(x+1)+2}$?
2) How is the graph ${\displaystyle g(x)=x^{3}+2}$ obtained from ${\displaystyle f(x)=x^{3}}$?
3) How is the graph ${\displaystyle g(x)=(x+1)^{2}}$ obtained from ${\displaystyle f(x)=x^{2}}$?
1) The basic graph is ${\displaystyle y=\log _{2}(x)}$.
2) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ up 2 unit.
3) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ to the left by 1 unit.

Solution:

Step 1:
We start with the basic graph of ${\displaystyle g(x)=\log _{2}(x)}$.
To get the graph of ${\displaystyle f(x)}$ from ${\displaystyle g(x)}$, we shift the graph of ${\displaystyle g(x)}$ up 2 and to the left by 1.
Step 2:
Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at ${\displaystyle x=-1}$.
Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at ${\displaystyle x=-1}$.