1. **State the problem:** We are given the function $f(x) = x^2 - 4x$ and want to analyze it. 2. **Formula and rules:** This is a quadratic function of the form $ax^2 + bx + c$ where $a=1$, $b=-4$, and $c=0$. 3. **Find the vertex:** The vertex of a parabola $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. 4. **Calculate vertex x-coordinate:** $$x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2$$ 5. **Calculate vertex y-coordinate:** $$f(2) = (2)^2 - 4(2) = 4 - 8 = -4$$ 6. **Vertex point:** The vertex is at $(2, -4)$. 7. **Find x-intercepts:** Set $f(x) = 0$: $$x^2 - 4x = 0$$ Factor: $$x(x - 4) = 0$$ So, $x=0$ or $x=4$. 8. **Find y-intercept:** Set $x=0$: $$f(0) = 0^2 - 4(0) = 0$$ So, y-intercept is at $(0,0)$. 9. **Summary:** The parabola opens upwards (since $a=1>0$), vertex at $(2,-4)$, x-intercepts at $(0,0)$ and $(4,0)$, and y-intercept at $(0,0)$. **Final answer:** The function $f(x) = x^2 - 4x$ has vertex at $(2,-4)$ and roots at $x=0$ and $x=4$.