1. **State the problem:** We want to understand how to solve or analyze the function $y=(x-1)^2 + 2$ on a graph. 2. **Formula and explanation:** This is a quadratic function in vertex form: $$y = (x-h)^2 + k$$ where $(h,k)$ is the vertex of the parabola. 3. **Identify the vertex:** Here, $h=1$ and $k=2$, so the vertex is at the point $(1,2)$. 4. **Shape and direction:** Since the coefficient of the squared term is positive (1), the parabola opens upwards. 5. **Axis of symmetry:** The vertical line $x=1$ is the axis of symmetry. 6. **Find y-intercept:** Set $x=0$, then $$y = (0-1)^2 + 2 = 1 + 2 = 3.$$ So the graph crosses the y-axis at $(0,3)$. 7. **Find x-intercepts:** Set $y=0$ and solve for $x$: $$0 = (x-1)^2 + 2$$ $$ (x-1)^2 = -2$$ Since the square of a real number cannot be negative, there are no real x-intercepts. 8. **Summary:** The parabola has vertex at $(1,2)$, opens upwards, crosses y-axis at $(0,3)$, and has no real x-intercepts. This analysis helps you graph the function and understand its behavior.