1. The problem is to understand the rules for graphing a quadratic equation. 2. A quadratic equation is generally written as $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$. 3. The graph of a quadratic equation is a parabola. The direction of the parabola depends on the sign of $a$: - If $a > 0$, the parabola opens upwards. - If $a < 0$, the parabola opens downwards. 4. The vertex of the parabola is the highest or lowest point, depending on the direction. It can be found using the formula: $$x = -\frac{b}{2a}$$ Substitute this $x$ value back into the equation to find the $y$ coordinate of the vertex. 5. The axis of symmetry is the vertical line that passes through the vertex, given by: $$x = -\frac{b}{2a}$$ 6. The y-intercept is the point where the graph crosses the y-axis, found by evaluating $y$ when $x=0$: $$y = c$$ 7. The x-intercepts (roots) are the points where the graph crosses the x-axis, found by solving: $$ax^2 + bx + c = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 8. Plot the vertex, axis of symmetry, y-intercept, and x-intercepts (if real) to sketch the parabola. 9. The parabola is symmetric about the axis of symmetry. 10. The width of the parabola depends on the absolute value of $a$: larger $|a|$ means narrower parabola, smaller $|a|$ means wider parabola.