📚 A-Level Maths: Quadratic & Cubic Curves Worksheet Essentials | A-Level 数学：二次与三次曲线知识点精讲
Quadratic and cubic curves are fundamental to A-Level Mathematics, appearing in algebra, coordinate geometry, and calculus. This worksheet-style revision article breaks down the essential concepts, graphing techniques, and common pitfalls, helping you build confidence and achieve higher marks in your exams.
二次曲线和三次曲线是 A-Level 数学的基础，贯穿代数、坐标几何和微积分。本工作纸式的知识点精讲将深入剖析核心概念、绘图技巧和常见错误，帮助你建立信心，在考试中取得更高分数。
1. Standard Form of a Quadratic | 二次函数的标准形式
A quadratic function can be written in standard form as f(x) = ax² + bx + c, where a, b and c are constants and a ≠ 0. The coefficient a determines the shape and orientation of the parabola, while c directly gives the y-intercept. The term bx affects the position of the vertex and the symmetry of the curve.
二次函数的标准形式为 f(x) = ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。系数 a 决定了抛物线的形状和开口方向，c 直接给出 y 轴截距。bx 项影响顶点的位置和曲线的对称性。
If a > 0, the graph is a minimum (∪-shaped); if a < 0, it is a maximum (∩-shaped). The magnitude of a controls the steepness: larger |a| makes the parabola narrower, while smaller |a| makes it wider.
若 a > 0，图像为最小值类型（∪ 形）；若 a < 0，则为最大值类型（∩ 形）。a 的绝对值控制陡峭程度：|a| 越大抛物线越窄，越小则越宽。
2. Graph of a Quadratic Curve (Parabola) | 二次曲线的图像（抛物线）
The graph of a quadratic function is called a parabola. It is symmetric about a vertical line known as the axis of symmetry. The point where the parabola turns is the vertex; it represents the maximum or minimum value of the function. Parabolas can intersect the x-axis at zero, one or two points, known as roots or x-intercepts.
二次函数的图像称为抛物线。它关于一条垂直线（对称轴）对称。抛物线转向的点是顶点，代表函数的最大值或最小值。抛物线与 x 轴可以交于 0 个、1 个或 2 个点，这些点称为根或 x 轴截距。
When a > 0, the parabola opens upwards, the vertex is a minimum, and the range is [y_min, ∞). When a < 0, the parabola opens downwards, the vertex is a maximum, and the range is (-∞, y_max].
当 a > 0 时，抛物线开口向上，顶点为最小值点，值域为 [y_min, ∞)；当 a < 0 时，开口向下，顶点为最大值点，值域为 (-∞, y_max]。
3. Finding the Roots of a Quadratic | 二次方程求根
Roots can be found by factorising, completing the square, or using the quadratic formula. The quadratic formula works for any quadratic equation ax² + bx + c = 0 and is given by:
求根可通过因式分解、配方法或求根公式完成。求根公式适用于任何二次方程 ax² + bx + c = 0，表达式如下：
x = [–b ± √(b² – 4ac)] / (2a)
Factorising is often quicker when the quadratic simplifies to (x – p)(x – q) = 0, giving roots x = p and x = q. Completing the square is particularly useful for deriving the vertex form and for solving when a ≠ 1.
当二次式可化简为 (x – p)(x – q) = 0 时，因式分解通常更快，根为 x = p 和 x = q。配方法在推导顶点形式以及 a ≠ 1 时求解特别有用。
Always check your roots by substituting them back into the original equation. A common mistake is losing a factor when dividing by x – make sure to keep all possible solutions.
求出根后务必代入原方程检验。常见错误是在除以 x 时丢失因子 —— 确保保留所有可能的解。
4. The Discriminant | 判别式
The discriminant of a quadratic equation ax² + bx + c = 0 is Δ = b² – 4ac. It tells us the nature and number of real roots without solving the equation completely.
二次方程 ax² + bx + c = 0 的判别式为 Δ = b² – 4ac。它可以告诉我们实根的性质和个数，而无需完全解出方程。
| Discriminant (Δ) | Nature of Roots / 根的情况 | Sketch clue / 草图线索 |
| Δ > 0 | Two distinct real roots / 两个不等实根 | Crosses x-axis at two points / 穿过 x 轴两点 |
| Δ = 0 | One repeated real root / 一个重根 | Touches x-axis at vertex / 顶点触碰 x 轴 |
| Δ < 0 | No real roots / 无实根 | Does not cut x-axis / 不与 x 轴相交 |
The discriminant is also helpful when trying to determine if a quadratic is always positive or always negative for all x – this requires a > 0 and Δ < 0 for always positive, or a < 0 and Δ < 0 for always negative.
判别式还可用于判断二次函数是否对所有 x 恒正或恒负 —— 恒正需要 a > 0 且 Δ < 0，恒负需要 a < 0 且 Δ < 0。
5. Vertex and Axis of Symmetry | 顶点与对称轴
The vertex of a parabola is the turning point. Using completing the square, we write f(x) = a(x – h)² + k, where the vertex is (h, k) and the axis of symmetry is the vertical line x = h. Alternatively, for the standard form, h = –b/(2a) and k = f(h).
抛物线的顶点是其转折点。通过配方法，可将函数写为 f(x) = a(x – h)² + k，此时顶点为 (h, k)，对称轴为垂直线 x = h。对于标准形式，可利用 h = –b/(2a) 和 k = f(h) 求得。
The vertex form is extremely useful for sketching, solving optimisation problems, and understanding transformations. For example, f(x) = 2(x + 3)² – 5 is a shift 3 units left, stretched vertically by factor 2, and moved down by 5.
顶点形式在草图绘制、求解优化问题和理解变换时极为有用。例如，f(x) = 2(x + 3)² – 5 表示向左平移 3 个单位、纵向拉伸为原来 2 倍并向下平移 5 个单位。
Vertex: (h, k) = (–b/(2a), f(–b/(2a)))
6. The y-Intercept and Symmetry | y 截距与对称性
The y-intercept of any quadratic y = ax² + bx + c is simply the constant term c. To find it, set x = 0, giving the point (0, c). This is always on the curve and is a quick check when sketching.
任何二次函数 y = ax² + bx + c 的 y 截距就是常数项 c。令 x = 0 即得点 (0, c)。该点总在曲线上，绘图时可快速用于校验。
Parabolas are symmetric about their axis of symmetry x = h. This means that if you know one point on one side of the parabola, you can find its mirror point on the other side. For example, if x = h + d gives y = f(h + d), then x = h – d gives the same y-value. This symmetry helps when plotting points.
抛物线关于对称轴 x = h 对称。若已知抛物线一侧的一个点，可找到其另一侧的镜像点。例如，若 x = h + d 给出 y = f(h + d)，则 x = h – d 给出相同的 y 值。这种对称性有助于描点作图。
7. Standard Form of a Cubic | 三次函数的标准形式
A cubic function has the general form f(x) = ax³ + bx² + cx + d, with a ≠ 0. The leading coefficient a dictates the end behaviour: for large |x|, the x³ term dominates. If a > 0, as x → –∞, y → –∞ and as x → +∞, y → +∞; if a < 0, the directions are reversed.
三次函数的一般形式为 f(x) = ax³ + bx² + cx + d, a ≠ 0。首项系数 a 决定末端走势：当 |x| 很大时，x³ 项起主导作用。若 a > 0，则 x → –∞ 时 y → –∞，x → +∞ 时 y → +∞；若 a < 0，方向相反。
Unlike quadratics, cubics are not symmetric about a vertical line, but they may have rotational symmetry about an inflection point. They can have up to two turning points and always have at least one real root.
与二次函数不同，三次函数没有关于垂直线的对称性，但可能关于某个拐点具有旋转对称性。三次曲线最多有两个转向点，且总是至少有一个实根。
8. Shapes of Cubic Curves | 三次曲线的图像形状
Cubic curves fall into several typical shapes depending on the coefficient a and the number of turning points. With a > 0, the basic shape rises from left to right; the curve may be strictly increasing (one real root, no turning points) or have a local maximum followed by a local minimum (three distinct real roots).
根据系数 a 和转向点个数，三次曲线可分为几类典型形状。a > 0 时基本形状从左到右上升；曲线可能严格单调递增（一个实根，无转向点）或具有一个局部极大值后跟一个局部极小值（三个不同的实根）。
When a < 0, the curve falls from left to right, again with possible zero or two turning points. A cubic with a repeated real root touches the x-axis but does not cross it at that point, creating a stationary point of inflection. Recognizing these shapes is essential for sketching.
当 a < 0 时，曲线从左到右下降，同样可能没有或有二个转向点。具有一个重复实根的三次曲线在相应点处触碰 x 轴但不穿过，形成一个驻点拐点。识别这些形状对草图绘制至关重要。
9. Factorising Cubics and Finding Roots | 三次函数的因式分解与求根
Factorising a cubic often starts by searching for an integer root using the factor theorem: if f(p) = 0, then (x – p) is a factor. After dividing by (x – p), we obtain a quadratic factor that can be solved to find the remaining roots.
三次函数的因式分解通常从寻找整数根开始，利用因式定理：若 f(p) = 0，则 (x – p) 是一个因式。除以 (x – p) 后得到一个二次因式，进一步求解可得其余根。
For example, to factorise x³ – 6x² + 11x – 6, test small values: f(1) = 1 – 6 + 11 – 6 = 0, so (x – 1) is a factor. Performing polynomial division yields (x – 1)(x² – 5x + 6) = (x – 1)(x – 2)(x – 3), giving roots x = 1, 2, 3.
例如，要分解 x³ – 6x² + 11x – 6，测试小值：f(1) = 1 – 6 + 11 – 6 = 0，所以 (x – 1) 是因式。多项式除法得到 (x – 1)(x² – 5x + 6) = (x – 1)(x – 2)(x – 3)，根为 x = 1, 2, 3。
Factor theorem: f(p) = 0 ⇔ (x – p) is a factor
10. Turning Points and Inflection Points of Cubics | 三次曲线的转向点与拐点
To find turning points, differentiate the cubic: f'(x) = 3ax² + 2bx + c. Setting f'(x) = 0 yields a quadratic, whose solutions give the x-coordinates of turning points. The discriminant of this derivative quadratic tells us how many turning points exist: if Δderiv > 0, there are two; if Δderiv = 0, there is one stationary inflection point; if Δderiv < 0, no turning points and the cubic is strictly monotonic.
要找转向点，对三次函数求导：f'(x) = 3ax² + 2bx + c。令 f'(x) = 0 得到一个二次方程，其解给出转向点的 x 坐标。此导数二次式的判别式告诉我们转向点的个数：若导数判别式 > 0，有两个转向点；若 = 0，有一个驻点拐点；若 < 0，无转向点，三次函数严格单调。
The inflection point occurs where the second derivative changes sign: f”(x) = 6ax + 2b = 0 gives x = –b/(3a). This point always lies midway between the two turning points (if they exist). The curvature changes from concave up to concave down, or vice versa, at the inflection point.
拐点出现在二阶导数变号之处：f”(x) = 6ax + 2b = 0 得 x = –b/(3a)。该点总是位于两个转向点（若存在）的正中间。在拐点处，曲线的凹凸性发生改变。
Inflection x = –b/(3a)
11. Sketching Quadratic and Cubic Curves | 二次与三次曲线草图绘制技巧
A systematic approach to sketching ensures accuracy and saves time. For quadratics, determine: (1) shape from a, (2) y-intercept c, (3) roots via factorising or formula, (4) vertex via completing square or –b/(2a), and (5) axis of symmetry. Plot these key points and draw a smooth symmetric curve.
采用系统化的方法绘制草图既可保证准确性又可节省时间。对于二次函数，按以下顺序确定：(1) 由 a 定开口方向，(2) y 截距 c，(3) 通过分解或公式求根，(4) 由配方法或 –b/(2a) 得顶点，(5) 对称轴。标出这些关键点，画出光滑对称的曲线。
For cubics, follow these steps: (1) end behaviour from sign of a, (2) y-intercept d, (3) find roots by factorising if possible, (4) find turning points via differentiation, (5) find inflection point x = –b/(3a). Mark all intercepts, turning points, and inflection; then sketch a smooth curve respecting the end behaviour and passing through all plotted points.
对于三次函数，按以下步骤：(1) 由 a 的正负定末端行为，(2) y 截距 d，(3) 尽可能通过因式分解求根，(4) 求导得转向点，(5) 求拐点 x = –b/(3a)。标出所有截距、转向点和拐点；然后绘制光滑曲线，确保符合末端行为并通过所有标出的点。
Always check that a cubic with three distinct real roots must have two turning points, and the sign of y values between roots should alternate. Labelling coordinates of key points on the sketch is often required in exam questions.
务必检验：具有三个不同实根的三次函数必有两个转向点，且各根之间的 y 值符号应交替变化。考试题目通常要求在草图上标出关键点的坐标。
12. Common Mistakes and Checkpoints | 常见错误与检查方法
One frequent error is forgetting that a quadratic may have no real roots, and mistakenly drawing it crossing the x-axis. Always check the discriminant first. With cubics, a common slip is ignoring the multiplicity of a root: a repeated root does not cross the axis but touches it, affecting the sketch locally.
常见错误之一是忘记二次函数可能无实根，而错误地画出其穿过 x 轴。务必先核查判别式。对于三次函数，容易忽略根的重数：重复根并不穿过 x 轴而是触碰，这会影响局部图形。
Another mistake is misplacing the vertex when using the formula x = –b/(2a) without calculating the corresponding y-value properly. Also, when factorising a cubic by grouping, ensure you take out the common factor correctly – double-check with expansion. Finally, when sketching, make sure the number of turning points matches the derivative’s discriminant, and never draw more turning points than the degree minus one allows.
另一个错误是使用公式 x = –b/(2a) 时未正确计算相应的 y 值导致顶点位置错误。此外，用分组法分解三次式时，确保正确提取公因子 —— 可通过展开检验。最后，草图绘制时要保证转向点个数与导数判别式一致，且画出的转向点数不得超过次数减一。
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