Abs Value Shift 3A861B

1. **State the problem:** We need to sketch the graph of the function $$y = |x - 2a|$$ where $$a$$ is a positive constant. 2. **Recall the absolute value function:** The graph of $$y = |x|$$ is a V-shaped graph with vertex at the origin $$(0,0)$$ and opens upward. 3. **Effect of the transformation:** The function $$y = |x - 2a|$$ shifts the graph of $$y = |x|$$ horizontally to the right by $$2a$$ units because of the term $$x - 2a$$ inside the absolute value. 4. **Vertex and axis of symmetry:** The vertex of the graph is at $$(2a, 0)$$ and the axis of symmetry is the vertical line $$x = 2a$$. 5. **Shape of the graph:** The graph is V-shaped, opening upward, with two linear pieces: - For $$x \\geq 2a$$, $$y = x - 2a$$ - For $$x < 2a$$, $$y = -(x - 2a) = 2a - x$$ 6. **Summary:** The graph is a V-shaped absolute value graph with vertex at $$(2a, 0)$$ and axis of symmetry $$x = 2a$$, shifted right by $$2a$$ units from the origin. **Final answer:** The graph of $$y = |x - 2a|$$ is a V-shaped graph opening upward with vertex at $$(2a, 0)$$ and axis of symmetry $$x = 2a$$.