Certainly! Let's analyze the given absolute value function step-by-step:
The given function is:
[tex]\[
y = -\frac{1}{2} |x - 1| + 2
\][/tex]
1. Vertex:
To find the vertex, we look at the inside of the absolute value and the constant term outside the absolute value.
The general form for an absolute value function is:
[tex]\[
y = a |x - h| + k
\][/tex]
where [tex]\((h, k)\)[/tex] is the vertex.
In our equation [tex]\(y = -\frac{1}{2} |x - 1| + 2\)[/tex], we identify [tex]\(h = 1\)[/tex] and [tex]\(k = 2\)[/tex]. Therefore, the vertex is:
[tex]\[
(1, 2)
\][/tex]
2. Opens:
The coefficient [tex]\(a\)[/tex] in the general form determines whether the graph opens up or down. If [tex]\(a\)[/tex] is positive, the graph opens up. If [tex]\(a\)[/tex] is negative, the graph opens down.
In our equation, [tex]\(a = -\frac{1}{2}\)[/tex], which is negative. So, the graph opens down.
3. Relation to Parent Function:
The parent function of an absolute value equation is [tex]\(y = |x|\)[/tex]. The coefficient [tex]\(a\)[/tex] also determines the width of the graph compared to the parent function.
- If [tex]\(|a| < 1\)[/tex], the graph is wider.
- If [tex]\(|a| > 1\)[/tex], the graph is narrower.
- If [tex]\(|a| = 1\)[/tex], the graph has the same width as the parent function.
In our equation, [tex]\(|a| = |\frac{1}{2}| = 0.5\)[/tex], which is less than 1. So, the graph is wider than the parent function.
4. Domain:
The domain of any absolute value function is all real numbers since [tex]\(x\)[/tex] can take any value. Therefore, the domain is:
[tex]\[
\text{All real numbers}
\][/tex]
5. Range:
The range of the function depends on the vertex and the direction in which the graph opens.
Because the graph opens down and the vertex is at [tex]\((1, 2)\)[/tex], the maximum value of the function is 2, and it decreases from there. Therefore, the range is:
[tex]\[
[-\infty, 2]
\][/tex]
### Summary:
- Vertex: [tex]\((1, 2)\)[/tex]
- Opens: Down
- Relation to Parent Function: Wider
- Domain: All real numbers
- Range: [tex]\([- \infty, 2]\)[/tex]
