005 Sample Final A, Question 17

Question Graph the following function,

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

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

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

Solution:

Step 1:
We start with the basic graph of $g(x)=\log _{2}(x)$.
To get the graph of $f(x)$ from $g(x)$, we shift the graph of $g(x)$ 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 $x=-1$.

Final Answer:
Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at $x=-1$.

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