005 Sample Final A, Question 18

Question Graph the following function,

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

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

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

Final Answer:
To get the graph of $f(x)$ from $\left({\frac {1}{3}}\right)^{x}$, we shift the graph of $\left({\frac {1}{3}}\right)^{x}$ up 1 and to the left by 1.

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