1. The problem is to analyze and understand the function $$f(x) = |x - 1| + 2$$ and describe its graph. 2. The absolute value function $$|x - 1|$$ represents the distance of $$x$$ from 1 on the number line, so it is always non-negative. 3. The function shifts the basic absolute value graph $$|x|$$ to the right by 1 unit (because of $$x - 1$$) and then shifts it up by 2 units (because of the $$+2$$). 4. To find key points, evaluate $$f(x)$$ at some values: - At $$x = 1$$: $$f(1) = |1 - 1| + 2 = 0 + 2 = 2$$ - At $$x = 0$$: $$f(0) = |0 - 1| + 2 = 1 + 2 = 3$$ - At $$x = 2$$: $$f(2) = |2 - 1| + 2 = 1 + 2 = 3$$ 5. The graph is a "V" shape with the vertex at $$(1, 2)$$. 6. The function is piecewise linear: $$ \begin{cases} f(x) = -(x - 1) + 2 = -x + 3 & \text{if } x < 1 \\ f(x) = (x - 1) + 2 = x + 1 & \text{if } x \geq 1 \end{cases} $$ 7. This means the left side of the vertex has slope $$-1$$ and the right side has slope $$1$$. 8. The graph crosses the y-axis at $$x=0$$ with $$f(0) = 3$$. Final answer: The graph of $$f(x) = |x - 1| + 2$$ is a "V" shaped graph with vertex at $$(1, 2)$$, opening upwards, with slopes $$-1$$ to the left and $$1$$ to the right.