Sure! Let's go through the steps to graph the function [tex]\( f(x) = \log_2(x - 1) + 2 \)[/tex].
### Step 1: Understand the Function
- The function [tex]\( \log_2(x - 1) \)[/tex] is defined only when [tex]\( x - 1 > 0 \)[/tex], which implies [tex]\( x > 1 \)[/tex].
- Therefore, the domain of the function is [tex]\( x > 1 \)[/tex].
- The function [tex]\( \log_2(x - 1) \)[/tex] indicates a logarithmic transformation shifted to the right by 1 unit.
- The additional [tex]\( +2 \)[/tex] outside the logarithm translates the graph 2 units upwards.
### Step 2: Identify Key Features
- Domain: [tex]\( x > 1 \)[/tex]
- Vertical Asymptote: At [tex]\( x = 1 \)[/tex], since the logarithmic part [tex]\( \log_2(x - 1) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- Horizontal Translation: The function is shifted to the right by 1 unit.
- Vertical Translation: The function is shifted up by 2 units.
### Step 3: Pick Values for [tex]\( x \)[/tex]
Choose significant values of [tex]\( x \)[/tex] greater than 1 to calculate [tex]\( f(x) \)[/tex]:
For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \log_2(2 - 1) + 2 = \log_2(1) + 2 = 0 + 2 = 2 \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = \log_2(4 - 1) + 2 = \log_2(3) + 2 \approx 1.58496 + 2 = 3.58496 \][/tex]
For [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = \log_2(8 - 1) + 2 = \log_2(7) + 2 \approx 2.80735 + 2 = 4.80735 \][/tex]
For [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = \log_2(10 - 1) + 2 = \log_2(9) + 2 \approx 3.16993 + 2 = 5.16993 \][/tex]
### Step 4: Plot the Points
Now, plot the points calculated above:
- [tex]\( (2, 2) \)[/tex]
- [tex]\( (4, 3.585) \)[/tex]
- [tex]\( (8, 4.807) \)[/tex]
- [tex]\( (10, 5.170) \)[/tex]
### Step 5: Draw the Graph
1. Draw a vertical asymptote at [tex]\( x = 1 \)[/tex].
2. Plot the points calculated.
3. Connect these points smoothly, extending the curve while ensuring it approaches the asymptote without touching it.
### Step 6: Label the Graph
- Label the x-axis and y-axis.
- Mark the vertical asymptote at [tex]\( x = 1 \)[/tex].
- Label a few key points such as [tex]\( (2, 2) \)[/tex], [tex]\( (4, 3.585) \)[/tex], etc.
- Add a title to the graph: "Graph of [tex]\( f(x) = \log_2(x - 1) + 2 \)[/tex]"
### Example Graph
The graph should appear as follows (this is a rough sketch based on the calculations):
```
y
|
5 |
4 |
3 |
2 | *
1 |
+--------------------------------------------
1 2 3 4 5 6 7 8 9 10 x
```
Note:
- As [tex]\( x \)[/tex] approaches 1 from the right, [tex]\( f(x) \)[/tex] drops sharply towards [tex]\( -\infty \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] continues to rise, but at a decreasing rate.
This is the graph of the logarithmic function [tex]\( f(x) = \log_2(x - 1) + 2 \)[/tex].
