The equation $y = \left\vert x - 2 \right\vert$ represents a piecewise linear function. The absolute value function affects the graph, creating a V-shape.

Breakdown of the function:

1. Definition:

   • The absolute value function $\left\vert x - 2 \right\vert$ is defined as:
   x - 2 & \text{if } x \geq 2 \\ -(x - 2) & \text{if } x < 2 \end{cases}$$ This means the function behaves differently on either side of $$x = 2$$.

2. Piecewise Form:

   • For $x \geq 2$: $y = x - 2$.
   • For $x < 2$: $y = -x + 2$.

3. Vertex:

   • The vertex of the graph occurs at the point where $x = 2$, giving $y = 0$. Thus, the vertex is at $(2, 0)$.

4. Graph Shape:

   • The graph consists of two linear segments:
     • A line with a positive slope $m = 1$ for $x \geq 2$.
     • A line with a negative slope $m = -1$ for $x < 2$.

5. Symmetry:

   • The graph is symmetric about the vertical line $x = 2$.

Graphical Representation:

• The graph is a V-shape opening upwards, with the vertex at $(2, 0)$.

Summary:

• The function increases as $x$ moves away from 2 in either direction, with the slope being positive on the right and negative on the left.

Would you like to see the graph of this function, or do you have any questions about the explanation?

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Here are 5 questions related to this function:

1. How would the graph change if the function was $y = \left\vert x + 2 \right\vert$?
2. What is the domain and range of the function $y = \left\vert x - 2 \right\vert$?
3. How does the vertex of the graph change if the function is $y = \left\vert x - 2 \right\vert + 3$?
4. What happens to the graph if the function becomes $y = 2\left\vert x - 2 \right\vert$?
5. Can you find the x-intercepts and y-intercepts for the function $y = \left\vert x - 2 \right\vert$?

Tip: Understanding how the absolute value function affects the graph can help you quickly sketch the graph of similar functions.