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!
