To find the vertex of the absolute value function [tex]\( y = |x| \)[/tex], let's go through the step-by-step process.
1. Understand the Absolute Value Function: The equation [tex]\( y = |x| \)[/tex] represents an absolute value function, which is a piecewise function defined as:
[tex]\[
y =
\begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}
\][/tex]
This function forms a V shape on the coordinate plane, with the point where the two lines meet being the vertex.
2. Find the Vertex: In general, the vertex of an absolute value function [tex]\( y = a|x - h| + k \)[/tex] is at the point [tex]\((h, k)\)[/tex]. For the basic function [tex]\( y = |x| \)[/tex]:
- There is no horizontal shift, so [tex]\( h = 0 \)[/tex].
- There is no vertical shift, so [tex]\( k = 0 \)[/tex].
Thus, the vertex is located at [tex]\((0, 0)\)[/tex].
3. Conclusion: The vertex of the function [tex]\( y = |x| \)[/tex] is at the origin.
Given the choices provided:
- [tex]\((1, -1)\)[/tex]
- [tex]\((0, 0)\)[/tex]
- [tex]\((0, -1)\)[/tex]
- [tex]\((0, 1)\)[/tex]
The correct answer is [tex]\((0, 0)\)[/tex].
