1. **State the problem:** We want to understand how to graph the function $f(x) = x^2 - 2x + 1$. 2. **Formula and rules:** This is a quadratic function of the form $f(x) = ax^2 + bx + c$ where $a=1$, $b=-2$, and $c=1$. 3. **Find the vertex:** The vertex of a parabola $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. Calculate: $$x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1$$ Find $y$ at $x=1$: $$f(1) = 1^2 - 2 \times 1 + 1 = 1 - 2 + 1 = 0$$ So the vertex is at $(1, 0)$. 4. **Determine the shape:** Since $a=1 > 0$, the parabola opens upwards. 5. **Find intercepts:** - **Y-intercept:** Set $x=0$: $$f(0) = 0^2 - 2 \times 0 + 1 = 1$$ So the y-intercept is $(0,1)$. - **X-intercept(s):** Solve $x^2 - 2x + 1 = 0$. This factors as: $$(x - 1)^2 = 0$$ So the only root is $x=1$, meaning the parabola touches the x-axis at $(1,0)$. 6. **Plot points:** Plot the vertex $(1,0)$, y-intercept $(0,1)$, and a few more points like $(2,1)$ to see the shape. 7. **Draw the parabola:** Connect the points smoothly forming a U-shaped curve opening upwards with vertex at $(1,0)$. This is how you get the graph of $f(x) = x^2 - 2x + 1$ as described.