Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points

Ask by Murray Robbins. in the United States

Mar 25,2025

Question

Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points

Ask by Murray Robbins. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

1. The function is given by  
   $$
   f(x)=3^x+2.
   $$

2. Start with the parent function  
   $$
   g(x)=3^x.
   $$
   The transformation is a vertical shift upward by 2 units.

3. To obtain the transformed function, add 2 to the output of $$ g(x) $$:  
   $$
   f(x)=g(x)+2=3^x+2.
   $$

4. The domain of $$ g(x)=3^x $$ is  
   $$
   (-\infty,\infty),
   $$
   and the vertical shift does not affect the domain. Thus, the domain of $$ f(x) $$ is  
   $$
   (-\infty,\infty).
   $$

5. The range of $$ g(x)=3^x $$ is  
   $$
   (0,\infty).
   $$
   Shifting up by 2 changes the range to  
   $$
   (2,\infty).
   $$

6. The horizontal asymptote for $$ g(x)=3^x $$ is $$ y=0 $$. After shifting up by 2, the horizontal asymptote becomes  
   $$
   y=2.
   $$

7. Since the function $$ 3^x $$ is increasing (because the base $$ 3 > 1 $$), the shifted function $$ 3^x+2 $$ is also increasing.

8. The points on the graph of $$ g(x)=3^x $$ are:  
   $$
   (-1,\frac{1}{3}), \quad (0,1), \quad (1,3).
   $$
   Applying the transformation (vertical shift by 2) gives the points on $$ f(x)=3^x+2 $$ as:  
   $$
   \left(-1,\frac{1}{3}+2\right)=\left(-1,\frac{7}{3}\right),
   $$
   $$
   (0,1+2)=(0,3),
   $$
   $$
   (1,3+2)=(1,5).
   $$
The transformed points are $$ \left(-1, \frac{7}{3}\right) $$, $$ (0, 3) $$, and $$ (1, 5) $$.

Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points

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Use transformations to help graph the function . Identify the domain, range, and horizontal asymptote. Determine whether the function is increasing or decreasing. The points , and transform to the points

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Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points