Answer :
Absolutely, let's go through the process step-by-step.
### Step 1: Understanding the Function [tex]\( y = |x| + 4 \)[/tex]
The function [tex]\( y = |x| + 4 \)[/tex] involves the absolute value of [tex]\( x \)[/tex], which means it takes different expressions for [tex]\( x \)[/tex] values less than 0 and greater than or equal to 0:
1. When [tex]\( x \geq 0 \)[/tex], [tex]\( |x| = x \)[/tex], so the function becomes [tex]\( y = x + 4 \)[/tex].
2. When [tex]\( x < 0 \)[/tex], [tex]\( |x| = -x \)[/tex], so the function becomes [tex]\( y = -x + 4 \)[/tex].
### Step 2: Plotting Points for [tex]\( y = |x| + 4 \)[/tex]
We need to plot points for both cases:
For [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x = 0 \implies y = 0 + 4 = 4 \)[/tex]
- [tex]\( x = 1 \implies y = 1 + 4 = 5 \)[/tex]
- [tex]\( x = 2 \implies y = 2 + 4 = 6 \)[/tex]
- [tex]\( x = 3 \implies y = 3 + 4 = 7 \)[/tex]
- [tex]\( x = 4 \implies y = 4 + 4 = 8 \)[/tex]
For [tex]\( x < 0 \)[/tex]:
- [tex]\( x = -1 \implies y = -(-1) + 4 = 5 \)[/tex]
- [tex]\( x = -2 \implies y = -(-2) + 4 = 6 \)[/tex]
- [tex]\( x = -3 \implies y = -(-3) + 4 = 7 \)[/tex]
- [tex]\( x = -4 \implies y = -(-4) + 4 = 8 \)[/tex]
- [tex]\( x = -5 \implies y = -(-5) + 4 = 9 \)[/tex]
### Step 3: Plot these Points
Create a plot with the [tex]\( x \)[/tex] and corresponding [tex]\( y \)[/tex] values.
### Step 4: Reflect Across the Line [tex]\( y = x \)[/tex]
To obtain the inverse function, we can reflect the graph across the line [tex]\( y = x \)[/tex]. Reflecting over [tex]\( y = x \)[/tex] involves swapping each [tex]\( (x, y) \)[/tex] coordinate to [tex]\( (y, x) \)[/tex].
Points after reflection:
- [tex]\( (0, 4) \rightarrow (4, 0) \)[/tex]
- [tex]\( (1, 5) \rightarrow (5, 1) \)[/tex]
- [tex]\( (2, 6) \rightarrow (6, 2) \)[/tex]
- [tex]\( (3, 7) \rightarrow (7, 3) \)[/tex]
- [tex]\( (4, 8) \rightarrow (8, 4) \)[/tex]
- [tex]\( (5, 9) \rightarrow (9, 5) \)[/tex]
- [tex]\( (-1, 5) \rightarrow (5, -1) \)[/tex]
- [tex]\( (-2, 6) \rightarrow (6, -2) \)[/tex]
- [tex]\( (-3, 7) \rightarrow (7, -3) \)[/tex]
- [tex]\( (-4, 8) \rightarrow (8, -4) \)[/tex]
- [tex]\( (-5, 9) \rightarrow (9, -5) \)[/tex]
These are now the points of the inverse function, and you can plot these on the graph.
### Step 5: Graphing
Using graph paper or graphing software, perform the following:
1. Plot the points for [tex]\( y = |x| + 4 \)[/tex].
2. Draw the line [tex]\( y = x \)[/tex] as a reference.
3. Plot the reflected points (inverse function).
The resulting graph will show the original function [tex]\( y = |x| + 4 \)[/tex] and its inverse reflected across the line [tex]\( y = x \)[/tex].
### Final Graph
1. The original function [tex]\( y = |x| + 4 \)[/tex] forms a V-shape starting from [tex]\( (0, 4) \)[/tex].
2. The inverse graph will be a reflected V-shape, showing points like [tex]\( (4, 0) \)[/tex], [tex]\( (5, 1) \)[/tex], and so on.
This methodological approach enables us to graph the function and its inverse manually without computational errors.
### Step 1: Understanding the Function [tex]\( y = |x| + 4 \)[/tex]
The function [tex]\( y = |x| + 4 \)[/tex] involves the absolute value of [tex]\( x \)[/tex], which means it takes different expressions for [tex]\( x \)[/tex] values less than 0 and greater than or equal to 0:
1. When [tex]\( x \geq 0 \)[/tex], [tex]\( |x| = x \)[/tex], so the function becomes [tex]\( y = x + 4 \)[/tex].
2. When [tex]\( x < 0 \)[/tex], [tex]\( |x| = -x \)[/tex], so the function becomes [tex]\( y = -x + 4 \)[/tex].
### Step 2: Plotting Points for [tex]\( y = |x| + 4 \)[/tex]
We need to plot points for both cases:
For [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x = 0 \implies y = 0 + 4 = 4 \)[/tex]
- [tex]\( x = 1 \implies y = 1 + 4 = 5 \)[/tex]
- [tex]\( x = 2 \implies y = 2 + 4 = 6 \)[/tex]
- [tex]\( x = 3 \implies y = 3 + 4 = 7 \)[/tex]
- [tex]\( x = 4 \implies y = 4 + 4 = 8 \)[/tex]
For [tex]\( x < 0 \)[/tex]:
- [tex]\( x = -1 \implies y = -(-1) + 4 = 5 \)[/tex]
- [tex]\( x = -2 \implies y = -(-2) + 4 = 6 \)[/tex]
- [tex]\( x = -3 \implies y = -(-3) + 4 = 7 \)[/tex]
- [tex]\( x = -4 \implies y = -(-4) + 4 = 8 \)[/tex]
- [tex]\( x = -5 \implies y = -(-5) + 4 = 9 \)[/tex]
### Step 3: Plot these Points
Create a plot with the [tex]\( x \)[/tex] and corresponding [tex]\( y \)[/tex] values.
### Step 4: Reflect Across the Line [tex]\( y = x \)[/tex]
To obtain the inverse function, we can reflect the graph across the line [tex]\( y = x \)[/tex]. Reflecting over [tex]\( y = x \)[/tex] involves swapping each [tex]\( (x, y) \)[/tex] coordinate to [tex]\( (y, x) \)[/tex].
Points after reflection:
- [tex]\( (0, 4) \rightarrow (4, 0) \)[/tex]
- [tex]\( (1, 5) \rightarrow (5, 1) \)[/tex]
- [tex]\( (2, 6) \rightarrow (6, 2) \)[/tex]
- [tex]\( (3, 7) \rightarrow (7, 3) \)[/tex]
- [tex]\( (4, 8) \rightarrow (8, 4) \)[/tex]
- [tex]\( (5, 9) \rightarrow (9, 5) \)[/tex]
- [tex]\( (-1, 5) \rightarrow (5, -1) \)[/tex]
- [tex]\( (-2, 6) \rightarrow (6, -2) \)[/tex]
- [tex]\( (-3, 7) \rightarrow (7, -3) \)[/tex]
- [tex]\( (-4, 8) \rightarrow (8, -4) \)[/tex]
- [tex]\( (-5, 9) \rightarrow (9, -5) \)[/tex]
These are now the points of the inverse function, and you can plot these on the graph.
### Step 5: Graphing
Using graph paper or graphing software, perform the following:
1. Plot the points for [tex]\( y = |x| + 4 \)[/tex].
2. Draw the line [tex]\( y = x \)[/tex] as a reference.
3. Plot the reflected points (inverse function).
The resulting graph will show the original function [tex]\( y = |x| + 4 \)[/tex] and its inverse reflected across the line [tex]\( y = x \)[/tex].
### Final Graph
1. The original function [tex]\( y = |x| + 4 \)[/tex] forms a V-shape starting from [tex]\( (0, 4) \)[/tex].
2. The inverse graph will be a reflected V-shape, showing points like [tex]\( (4, 0) \)[/tex], [tex]\( (5, 1) \)[/tex], and so on.
This methodological approach enables us to graph the function and its inverse manually without computational errors.
