# 005 Sample Final A, Question 18

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 ${\displaystyle f(x)=3^{(x+1)}-2}$?
2) How is the graph ${\displaystyle g(x)=x+1}$ obtained from ${\displaystyle f(x)=x}$?
3) How is the graph ${\displaystyle g(x)=(x-3)^{2}}$ obtained from ${\displaystyle f(x)=x^{2}}$?
1) The basic graph is ${\displaystyle y=3^{x}}$.
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 right by 3 units.

Solution:

Step 1:
We start with the basic graph of ${\displaystyle g(x)=3^{x}}$.
To get the graph of ${\displaystyle f(x)}$ from ${\displaystyle g(x)}$, we shift the graph of ${\displaystyle g(x)}$ down 2 and to the left by 1.
Step 2:
Two ordered pairs are (-1,-1), and (0,1). There is a horizontal asymptote at ${\displaystyle y=-2}$.
To get the graph of ${\displaystyle f(x)}$ from ${\displaystyle 3^{x}}$, we shift the graph of ${\displaystyle 3^{x}}$ down 2 and to the left by 1.
Two ordered pairs are (-1,-1), and (0,1). There is a horizontal asymptote at ${\displaystyle y=-2}$