Sure! Let's graph the function [tex]\( f(x) = 3^{x+1} \)[/tex] step by step.
### Understanding the function
The function [tex]\( f(x) = 3^{x+1} \)[/tex] is an exponential function with a base of 3. This means that the value of the function will change exponentially based on the value of [tex]\( x \)[/tex].
### Identifying key points
To graph the function, we need to identify some key points. Let's calculate the value of [tex]\( f(x) \)[/tex] for different values of [tex]\( x \)[/tex]:
1. At [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3^{0+1} = 3^1 = 3
\][/tex]
2. At [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3^{1+1} = 3^2 = 9
\][/tex]
3. At [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3^{-1+1} = 3^0 = 1
\][/tex]
4. At [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 3^{-2+1} = 3^{-1} = \frac{1}{3}
\][/tex]
5. At [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 3^{2+1} = 3^3 = 27
\][/tex]
Now we have the following key points:
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (1, 9) \)[/tex]
- [tex]\( (-1, 1) \)[/tex]
- [tex]\( (-2, \frac{1}{3}) \)[/tex]
- [tex]\( (2, 27) \)[/tex]
### Plotting the points and sketching the graph
1. Start by plotting these points on a coordinate plane:
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (1, 9) \)[/tex]
- [tex]\( (-1, 1) \)[/tex]
- [tex]\( (-2, \frac{1}{3}) \)[/tex]
- [tex]\( (2, 27) \)[/tex]
2. Draw a smooth curve through these points to represent the graph of the function. The exponential function [tex]\( 3^{x+1} \)[/tex] will increase rapidly as [tex]\( x \)[/tex] increases, and will approach the x-axis (but never touch it) as [tex]\( x \)[/tex] decreases.
### Graph characteristics
- The domain of [tex]\( f(x) \)[/tex] is all real numbers [tex]\( (-\infty, \infty) \)[/tex].
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex] because the exponential function never reaches zero.
- The y-intercept is at point [tex]\( (0, 3) \)[/tex].
- The graph is increasing and continuous.
### Visual Representation
Here is a visual representation of the graph based on the calculated points:
```
y
↑
|
|
|
| (2, 27)
| /
| /
| /
| /
| /
| (1, 9) /
| / /
| / /
| (-2, 1/3) / /
| | / /
| | / /
| | / (0, 3)
| | /
| | /
| | /
| |/
|------|-----------|-----------|--------→
```
By plotting the points and drawing a smooth curve through them, you can clearly see the exponential growth behavior of the function [tex]\( f(x) = 3^{x+1} \)[/tex].
