1. **Problem statement:** Find the minimum point of the curve $y = x^2 - 2x - 4$. 2. **Formula and rules:** The minimum or maximum point of a quadratic function $y = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. 3. **Calculate the x-coordinate of the minimum:** Here, $a = 1$, $b = -2$, so $$x = -\frac{-2}{2 \times 1} = \frac{2}{2} = 1.$$ 4. **Calculate the y-coordinate of the minimum:** Substitute $x=1$ into the function: $$y = 1^2 - 2 \times 1 - 4 = 1 - 2 - 4 = -5.$$ 5. **Minimum point coordinates:** The minimum point is at $(1, -5)$. 1. **Problem statement:** Estimate the solutions to $x^2 - 2x - 4 = 0$ from the graph. 2. **Explanation:** The solutions are the x-values where the graph crosses the x-axis. 3. **From the graph:** The roots are approximately $-1.2$ and $3.2$ (to 1 decimal place). 1. **Problem statement:** Find the minimum point coordinates of $y = x^2 - 2x - 4$. 2. **Answer:** Minimum point: $(1, -5)$. 3. **Problem statement:** Estimate the solutions to $x^2 - 2x - 4 = 0$. 4. **Answer:** Solutions: $-1.2$, $3.2$ (to 1 decimal place). "slug": "quadratic curve", "subject": "algebra", "desmos": {"latex": "y=x^2-2x-4","features": {"intercepts": true,"extrema": true}}, "q_count": 3