Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
