1. **Problem Statement:** Sketch the function $f(x) = (x - 1)^2 + 2$ using transformations and find its domain and range. 2. **Formula and Rules:** The base function is $y = x^2$, a parabola with vertex at $(0,0)$. Transformations applied: - Horizontal shift right by 1: $x \to x-1$ - Vertical shift up by 2: $y \to y+2$ 3. **Step-by-step Transformation:** - Start with $y = x^2$. - Apply horizontal shift: $y = (x-1)^2$ moves the vertex from $(0,0)$ to $(1,0)$. - Apply vertical shift: $y = (x-1)^2 + 2$ moves vertex to $(1,2)$. 4. **Domain:** The parabola extends infinitely left and right, so domain is all real numbers: $$\text{Domain} = (-\infty, \infty)$$ 5. **Range:** Since $(x-1)^2 \geq 0$, minimum value of $f(x)$ is at vertex: $$f(1) = (1-1)^2 + 2 = 2$$ Thus, range is: $$\text{Range} = [2, \infty)$$ **Final answer:** - Domain: $(-\infty, \infty)$ - Range: $[2, \infty)$