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

Solution:

Step 1:
We start with the basic graph of $g(x)=3^{x}$.
To get the graph of $f(x)$ from $g(x)$, we shift the graph of $g(x)$ 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 $y=-2$.

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

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