Answer :
Sure, let's graph the function [tex]\( g(x) = -2|x-1| + 3 \)[/tex] based on [tex]\( f(x) = |x| \)[/tex]. To do this, we will go through the transformations step by step.
### Step-by-step Transformation:
1. Starting with the basic absolute value function:
[tex]\[ f(x) = |x| \][/tex]
2. Horizontal Shift:
The function [tex]\( |x-1| \)[/tex] indicates a horizontal shift to the right by 1 unit.
[tex]\[ f(x-1) = |x-1| \][/tex]
3. Vertical Stretch and Reflection:
The function [tex]\( -2|x-1| \)[/tex] involves two transformations.
- The factor of -2 indicates a vertical stretch by a factor of 2 and a reflection about the x-axis.
[tex]\[ g_1(x) = -2|x-1| \][/tex]
4. Vertical Shift:
Finally, adding 3 to the function shifts the graph upwards by 3 units.
[tex]\[ g(x) = -2|x-1| + 3 \][/tex]
### Analyzing the Function
Let's break this down into two parts (left and right of the vertex):
- For [tex]\( x \geq 1 \)[/tex]:
[tex]\[ g(x) = -2(x - 1) + 3 = -2x + 2 + 3 = -2x + 5 \][/tex]
- For [tex]\( x < 1 \)[/tex]:
[tex]\[ g(x) = -2(1 - x) + 3 = -2 + 2x + 3 = 2x + 1 \][/tex]
### Vertex of the Function
The vertex of the transformed function occurs at [tex]\( x = 1 \)[/tex]. At this point:
[tex]\[ g(1) = -2|1-1| + 3 = -2 \cdot 0 + 3 = 3 \][/tex]
So the vertex of [tex]\( g(x) \)[/tex] is [tex]\( (1, 3) \)[/tex].
### Graphical Representation
1. Vertex: The vertex of [tex]\( g(x) \)[/tex] is at [tex]\( (1, 3) \)[/tex].
2. Left Part ( [tex]\( x < 1 \)[/tex] ):
- The slope is [tex]\(\ 2 \)[/tex] and it passes through the vertex (1, 3).
- Equation: [tex]\(\ y = 2x + 1 \)[/tex]
3. Right Part ([tex]\( x \geq 1 \)[/tex]):
- The slope is [tex]\(\ -2 \)[/tex] and it also passes through the vertex (1, 3).
- Equation: [tex]\(\ y = -2x + 5 \)[/tex]
### Plotting the Graph
To plot these functions, follow these steps:
1. Draw a coordinate plane and mark the vertex at point [tex]\( (1, 3) \)[/tex].
2. For [tex]\( x < 1 \)[/tex] (left part): Draw a line starting from the vertex [tex]\( (1, 3) \)[/tex] with a slope of 2 (going upwards to the left).
3. For [tex]\( x \geq 1 \)[/tex] (right part): Draw a line starting from the vertex [tex]\( (1, 3) \)[/tex] with a slope of -2 (going downwards to the right).
This will create a 'V' shape that is inverted and shifted as described.
### Summary
The graph of [tex]\( g(x) = -2|x-1| + 3 \)[/tex] is an inverted V-shape with the vertex at [tex]\( (1, 3) \)[/tex], with the left arm having a slope of 2 and the right arm having a slope of -2.
### Step-by-step Transformation:
1. Starting with the basic absolute value function:
[tex]\[ f(x) = |x| \][/tex]
2. Horizontal Shift:
The function [tex]\( |x-1| \)[/tex] indicates a horizontal shift to the right by 1 unit.
[tex]\[ f(x-1) = |x-1| \][/tex]
3. Vertical Stretch and Reflection:
The function [tex]\( -2|x-1| \)[/tex] involves two transformations.
- The factor of -2 indicates a vertical stretch by a factor of 2 and a reflection about the x-axis.
[tex]\[ g_1(x) = -2|x-1| \][/tex]
4. Vertical Shift:
Finally, adding 3 to the function shifts the graph upwards by 3 units.
[tex]\[ g(x) = -2|x-1| + 3 \][/tex]
### Analyzing the Function
Let's break this down into two parts (left and right of the vertex):
- For [tex]\( x \geq 1 \)[/tex]:
[tex]\[ g(x) = -2(x - 1) + 3 = -2x + 2 + 3 = -2x + 5 \][/tex]
- For [tex]\( x < 1 \)[/tex]:
[tex]\[ g(x) = -2(1 - x) + 3 = -2 + 2x + 3 = 2x + 1 \][/tex]
### Vertex of the Function
The vertex of the transformed function occurs at [tex]\( x = 1 \)[/tex]. At this point:
[tex]\[ g(1) = -2|1-1| + 3 = -2 \cdot 0 + 3 = 3 \][/tex]
So the vertex of [tex]\( g(x) \)[/tex] is [tex]\( (1, 3) \)[/tex].
### Graphical Representation
1. Vertex: The vertex of [tex]\( g(x) \)[/tex] is at [tex]\( (1, 3) \)[/tex].
2. Left Part ( [tex]\( x < 1 \)[/tex] ):
- The slope is [tex]\(\ 2 \)[/tex] and it passes through the vertex (1, 3).
- Equation: [tex]\(\ y = 2x + 1 \)[/tex]
3. Right Part ([tex]\( x \geq 1 \)[/tex]):
- The slope is [tex]\(\ -2 \)[/tex] and it also passes through the vertex (1, 3).
- Equation: [tex]\(\ y = -2x + 5 \)[/tex]
### Plotting the Graph
To plot these functions, follow these steps:
1. Draw a coordinate plane and mark the vertex at point [tex]\( (1, 3) \)[/tex].
2. For [tex]\( x < 1 \)[/tex] (left part): Draw a line starting from the vertex [tex]\( (1, 3) \)[/tex] with a slope of 2 (going upwards to the left).
3. For [tex]\( x \geq 1 \)[/tex] (right part): Draw a line starting from the vertex [tex]\( (1, 3) \)[/tex] with a slope of -2 (going downwards to the right).
This will create a 'V' shape that is inverted and shifted as described.
### Summary
The graph of [tex]\( g(x) = -2|x-1| + 3 \)[/tex] is an inverted V-shape with the vertex at [tex]\( (1, 3) \)[/tex], with the left arm having a slope of 2 and the right arm having a slope of -2.
