005 Sample Final A, Question 18

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Question Graph the following function,

${\displaystyle f(x)=\left({\frac {1}{3}}\right)^{x+1}+1}$

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

Foundations
1) What is the basic graph of ${\displaystyle f(x)=\left({\frac {1}{3}}\right)^{x+1}+1}$?
2) How is the graph ${\displaystyle g(x)=x^{3}+1}$ 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 ${\displaystyle y=\left({\frac {1}{3}}\right)^{x}}$.
2) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ up 1 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 ${\displaystyle g(x)=\left({\frac {1}{3}}\right)^{x}}$.
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 ordered pairs are ${\displaystyle \left(0,{\frac {4}{3}}\right)}$   and   ${\displaystyle (-1,1)}$. There is a horizontal asymptote at ${\displaystyle y=1}$.
To get the graph of ${\displaystyle f(x)}$ from ${\displaystyle \left({\frac {1}{3}}\right)^{x}}$, we shift the graph of ${\displaystyle \left({\frac {1}{3}}\right)^{x}}$ up 1 and to the left by 1.