Answer :
Sure! Let's go through this process step by step to generate the graph of the function [tex]\( f(x) = |x - 2| - 4 \)[/tex].
### Step 1: Identify the Translation
We start with the parent function [tex]\( f(x) = |x| \)[/tex].
The given function [tex]\( f(x) = |x - 2| - 4 \)[/tex] involves two transformations:
1. Horizontal Translation: The term [tex]\( |x - 2| \)[/tex] means we shift the parent function [tex]\( |x| \)[/tex] 2 units to the right.
2. Vertical Translation: The term [tex]\(- 4\)[/tex] indicates that we shift the function 4 units down.
So, the function [tex]\( f(x) = |x - 2| - 4 \)[/tex] is the parent function [tex]\( |x| \)[/tex] translated 2 units to the right and 4 units down.
### Step 2: Create a Table of Values
To graph this function, let's create a table of values with selected [tex]\( x \)[/tex]-values and their corresponding [tex]\( y \)[/tex]-values using the function [tex]\( f(x) \)[/tex].
Here is the table of values for different [tex]\( x \)[/tex]-values:
| [tex]\( x \)[/tex] | [tex]\( y = |x - 2| - 4 \)[/tex] |
|----------|----------------------|
| [tex]\(-10\)[/tex] | 8 |
| [tex]\(-9\)[/tex] | 7 |
| [tex]\(-8\)[/tex] | 6 |
| [tex]\(-7\)[/tex] | 5 |
| [tex]\(-6\)[/tex] | 4 |
| [tex]\(-5\)[/tex] | 3 |
| [tex]\(-4\)[/tex] | 2 |
| [tex]\(-3\)[/tex] | 1 |
| [tex]\(-2\)[/tex] | 0 |
| [tex]\(-1\)[/tex] | -1 |
| [tex]\( 0 \)[/tex] | -2 |
| [tex]\( 1 \)[/tex] | -3 |
| [tex]\( 2 \)[/tex] | -4 |
| [tex]\( 3 \)[/tex] | -3 |
| [tex]\( 4 \)[/tex] | -2 |
| [tex]\( 5 \)[/tex] | -1 |
| [tex]\( 6 \)[/tex] | 0 |
| [tex]\( 7 \)[/tex] | 1 |
| [tex]\( 8 \)[/tex] | 2 |
| [tex]\( 9 \)[/tex] | 3 |
| [tex]\(10\)[/tex] | 4 |
### Step 3: Plot the Points and Draw the Graph
Using this table of values, you can plot the points on a coordinate plane and connect them to form the graph.
The graph will show a V shape, characteristic of absolute value functions, with its vertex (the lowest point) at the point (2, -4) due to the translations. From there, the lines will extend upwards to the left and to the right.
### Conclusion
We have identified the translations and used them to alter the parent function [tex]\( |x| \)[/tex]. We then created a table of values using the transformed function and plotted these points to form the graph of [tex]\( f(x) = |x - 2| - 4 \)[/tex].
### Step 1: Identify the Translation
We start with the parent function [tex]\( f(x) = |x| \)[/tex].
The given function [tex]\( f(x) = |x - 2| - 4 \)[/tex] involves two transformations:
1. Horizontal Translation: The term [tex]\( |x - 2| \)[/tex] means we shift the parent function [tex]\( |x| \)[/tex] 2 units to the right.
2. Vertical Translation: The term [tex]\(- 4\)[/tex] indicates that we shift the function 4 units down.
So, the function [tex]\( f(x) = |x - 2| - 4 \)[/tex] is the parent function [tex]\( |x| \)[/tex] translated 2 units to the right and 4 units down.
### Step 2: Create a Table of Values
To graph this function, let's create a table of values with selected [tex]\( x \)[/tex]-values and their corresponding [tex]\( y \)[/tex]-values using the function [tex]\( f(x) \)[/tex].
Here is the table of values for different [tex]\( x \)[/tex]-values:
| [tex]\( x \)[/tex] | [tex]\( y = |x - 2| - 4 \)[/tex] |
|----------|----------------------|
| [tex]\(-10\)[/tex] | 8 |
| [tex]\(-9\)[/tex] | 7 |
| [tex]\(-8\)[/tex] | 6 |
| [tex]\(-7\)[/tex] | 5 |
| [tex]\(-6\)[/tex] | 4 |
| [tex]\(-5\)[/tex] | 3 |
| [tex]\(-4\)[/tex] | 2 |
| [tex]\(-3\)[/tex] | 1 |
| [tex]\(-2\)[/tex] | 0 |
| [tex]\(-1\)[/tex] | -1 |
| [tex]\( 0 \)[/tex] | -2 |
| [tex]\( 1 \)[/tex] | -3 |
| [tex]\( 2 \)[/tex] | -4 |
| [tex]\( 3 \)[/tex] | -3 |
| [tex]\( 4 \)[/tex] | -2 |
| [tex]\( 5 \)[/tex] | -1 |
| [tex]\( 6 \)[/tex] | 0 |
| [tex]\( 7 \)[/tex] | 1 |
| [tex]\( 8 \)[/tex] | 2 |
| [tex]\( 9 \)[/tex] | 3 |
| [tex]\(10\)[/tex] | 4 |
### Step 3: Plot the Points and Draw the Graph
Using this table of values, you can plot the points on a coordinate plane and connect them to form the graph.
The graph will show a V shape, characteristic of absolute value functions, with its vertex (the lowest point) at the point (2, -4) due to the translations. From there, the lines will extend upwards to the left and to the right.
### Conclusion
We have identified the translations and used them to alter the parent function [tex]\( |x| \)[/tex]. We then created a table of values using the transformed function and plotted these points to form the graph of [tex]\( f(x) = |x - 2| - 4 \)[/tex].
