At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's analyze the function [tex]\( y = 3|x-1| + 1 \)[/tex] step by step.
### Step-by-Step Solution:
1. Identify the Vertex:
The vertex of an absolute value function [tex]\( y = a |x - h| + k \)[/tex] is given by the point [tex]\( (h, k) \)[/tex]. In this function, [tex]\( a = 3 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 1 \)[/tex]. Therefore, the vertex is:
[tex]\[ \text{Vertex: } (1, 1) \][/tex]
2. Determine if the Function Opens Up or Down:
The coefficient [tex]\( a \)[/tex] in front of the absolute value determines whether the function opens up or down. If [tex]\( a \)[/tex] is positive, the function opens upwards; if [tex]\( a \)[/tex] is negative, it opens downwards. Here, [tex]\( a = 3 \)[/tex], which is positive, so the function opens upwards.
[tex]\[ \text{Opens: Up} \][/tex]
3. Relation to Parent Function [tex]\( y = |x| \)[/tex]:
The parent function [tex]\( y = |x| \)[/tex] has been transformed as follows in the given function [tex]\( y = 3|x - 1| + 1 \)[/tex]:
- Vertical Stretch: The coefficient 3 causes a vertical stretch by a factor of 3.
- Horizontal Shift: The function is shifted to the right by 1 unit.
- Vertical Shift: The function is shifted up by 1 unit.
[tex]\[ \text{Relation to Parent Function: Vertical stretch by 3, shifted right by 1, and up by 1} \][/tex]
4. Determine the Domain:
The domain of any absolute value function [tex]\( y = a|x - h| + k \)[/tex] is all real numbers because there are no restrictions on the values that [tex]\( x \)[/tex] can take.
[tex]\[ \text{Domain: All Real Numbers} \][/tex]
5. Determine the Range:
Since the function opens upwards and the vertex is the lowest point on the graph at [tex]\( (1, 1) \)[/tex], the range includes all [tex]\( y \)[/tex] values greater than or equal to the y-coordinate of the vertex.
[tex]\[ \text{Range: } y \geq 1 \][/tex]
To summarize:
Vertex: [tex]\((1, 1)\)[/tex]
Opens: Up
Relation to Parent Function: Vertical stretch by 3, shifted right by 1, and up by 1
Domain: All Real Numbers
Range: [tex]\(y \geq 1\)[/tex]
Everything checks out!
### Step-by-Step Solution:
1. Identify the Vertex:
The vertex of an absolute value function [tex]\( y = a |x - h| + k \)[/tex] is given by the point [tex]\( (h, k) \)[/tex]. In this function, [tex]\( a = 3 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 1 \)[/tex]. Therefore, the vertex is:
[tex]\[ \text{Vertex: } (1, 1) \][/tex]
2. Determine if the Function Opens Up or Down:
The coefficient [tex]\( a \)[/tex] in front of the absolute value determines whether the function opens up or down. If [tex]\( a \)[/tex] is positive, the function opens upwards; if [tex]\( a \)[/tex] is negative, it opens downwards. Here, [tex]\( a = 3 \)[/tex], which is positive, so the function opens upwards.
[tex]\[ \text{Opens: Up} \][/tex]
3. Relation to Parent Function [tex]\( y = |x| \)[/tex]:
The parent function [tex]\( y = |x| \)[/tex] has been transformed as follows in the given function [tex]\( y = 3|x - 1| + 1 \)[/tex]:
- Vertical Stretch: The coefficient 3 causes a vertical stretch by a factor of 3.
- Horizontal Shift: The function is shifted to the right by 1 unit.
- Vertical Shift: The function is shifted up by 1 unit.
[tex]\[ \text{Relation to Parent Function: Vertical stretch by 3, shifted right by 1, and up by 1} \][/tex]
4. Determine the Domain:
The domain of any absolute value function [tex]\( y = a|x - h| + k \)[/tex] is all real numbers because there are no restrictions on the values that [tex]\( x \)[/tex] can take.
[tex]\[ \text{Domain: All Real Numbers} \][/tex]
5. Determine the Range:
Since the function opens upwards and the vertex is the lowest point on the graph at [tex]\( (1, 1) \)[/tex], the range includes all [tex]\( y \)[/tex] values greater than or equal to the y-coordinate of the vertex.
[tex]\[ \text{Range: } y \geq 1 \][/tex]
To summarize:
Vertex: [tex]\((1, 1)\)[/tex]
Opens: Up
Relation to Parent Function: Vertical stretch by 3, shifted right by 1, and up by 1
Domain: All Real Numbers
Range: [tex]\(y \geq 1\)[/tex]
Everything checks out!
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.