Certainly! Let's analyze the absolute value function [tex]\( y = 3|x - 11| + 1 \)[/tex].
1. Finding the Vertex:
The function given is in the form [tex]\( y = a|x - h| + k \)[/tex], which is the vertex form of an absolute value function.
Here, [tex]\( a = 3 \)[/tex], [tex]\( h = 11 \)[/tex], and [tex]\( k = 1 \)[/tex].
The vertex of the function is the point [tex]\((h, k)\)[/tex].
Vertex: [tex]\((11, 1)\)[/tex]
2. Opening Direction:
The value of 'a' (which is 3 in this case) determines whether the function opens up or down.
- If [tex]\( a > 0 \)[/tex], the function opens upward.
- If [tex]\( a < 0 \)[/tex], the function opens downward.
Since [tex]\( a = 3 \)[/tex] and [tex]\( 3 > 0 \)[/tex], the function opens up.
Opens: Up
3. Relation to the Parent Function [tex]\( y = |x| \)[/tex]:
The parent function for absolute value is [tex]\( y = |x| \)[/tex].
The given function has a coefficient [tex]\( |a| = 3 \)[/tex], where [tex]\( a \)[/tex] affects the width of the graph.
- If [tex]\( |a| > 1 \)[/tex], the graph is narrower than the parent function.
- If [tex]\( |a| < 1 \)[/tex], the graph is wider than the parent function.
Since [tex]\( |3| > 1 \)[/tex], the function is narrower than [tex]\( y = |x| \)[/tex].
Relation to Parent Function: Narrower
4. Domain:
The domain of any absolute value function is all real numbers because you can insert any real number for [tex]\( x \)[/tex] and get a corresponding [tex]\( y \)[/tex] value.
Domain: All real numbers
5. Range:
To find the range, consider the vertex and the direction the function opens.
- Since the function opens upward and the vertex is [tex]\((11, 1)\)[/tex], the smallest value [tex]\( y \)[/tex] can take is 1, and it can get larger from there.
Range: [tex]\( y \geq 1 \)[/tex]
In summary:
```
Vertex: (11, 1)
Opens: Up
Relation to Parent Function: Narrower
Domain: All real numbers
Range: y ≥ 1
```
