📚 A-Level Mathematics: Graphs and Key Features of Quartic Functions | A-Level 数学：四次函数图像与关键特征
A quartic function is a polynomial of degree four. Its graph is a smooth, continuous curve that can take many shapes, ranging from a single "W" or "M" form to more complex patterns. Understanding its key features — roots, turning points, symmetry and end behaviour — is essential for sketching and solving problems in A-Level Mathematics.
四次函数是四次多项式，其图像是光滑连续的曲线，形态多样，从简单的“W”形或“M”形到更复杂的模式。掌握它的关键特征——零点、驻点、对称性和端部行为——是A-Level数学中绘制图像和解题的关键。
1. Definition and General Form | 定义与一般形式
A quartic function can be written in the general form:
f(x) = ax⁴ + bx³ + cx² + dx + e, where a ≠ 0
The highest power is 4, so the degree is 4. The coefficients a, b, c, d are real numbers, and e is the constant term. The leading coefficient a determines the overall shape and the end behaviour of the graph.
四次函数的一般形式可写为：
f(x) = ax⁴ + bx³ + cx² + dx + e，其中 a ≠ 0
最高次项为4，因此次数是4。系数 a、b、c、d 为实数，e 为常数项。首项系数 a 决定了图像的整体形状和端部行为。
2. Leading Coefficient and End Behaviour | 首项系数与端部行为
For large positive or negative values of x, the quartic term ax⁴ dominates the expression. Because x⁴ is always positive (or zero), the sign of a controls the behaviour at both ends of the graph:
当 x 取绝对值很大的正值或负值时，四次项 ax⁴ 主导整个表达式。由于 x⁴ 始终非负，a 的符号便决定了图像两端的走向：
If a > 0, then f(x) → +∞ as x → ±∞. Both ends point upward.
若 a > 0，则当 x → ±∞ 时，f(x) → +∞，两端均向上延伸。
If a < 0, then f(x) → −∞ as x → ±∞. Both ends point downward.
若 a < 0，则当 x → ±∞ 时，f(x) → −∞，两端均向下延伸。
This is a key difference from cubic functions, whose ends go in opposite directions. For a quartic, both ends always go the same way.
这是与三次函数的重要区别：三次函数的两端方向相反，而四次函数的两端方向始终相同。
3. Roots and Factorised Form | 零点与因式形式
If a quartic function has real roots r₁, r₂, r₃ and r₄, it can be written in factorised form:
f(x) = a(x − r₁)(x − r₂)(x − r₃)(x − r₄)
ple, (x − r)² means the graph touches the axis at x = r and turns there, without crossing.
若四次函数有实数零点 r₁、r₂、r₃、r₄，则可写成因式形式：
f(x) = a(x − r₁)(x − r₂)(x − r₃)(x − r₄)
每个因式对应一个 x 轴交点。重根会产生与 x 轴相切的效果。例如 (x − r)² 表示图像在 x = r 处与轴相切并折回，而非穿过。
Four distinct real roots: the graph crosses the x-axis four times.
四个互异实根：图像与 x 轴相交四次。
One repeated root (double root) and two distinct roots: the graph touches once and crosses twice.
一个二重根加两个单根：图像相切一次、穿过两次。
Two double roots: the graph touches the x-axis at two points.
两个二重根：图像与 x 轴相切于两点。
No real roots: the graph never crosses the x-axis; it lies entirely above or below it (depending on a).
无实根：图像不与 x 轴相交，完全位于 x 轴上方或下方（取决于 a 的符号）。
4. Symmetry and Special Cases | 对称性与特殊情形
When the quartic is made up only of even powers, it is an even function. The general form becomes:
f(x) = ax⁴ + cx² + e
hs of equations such as y = x⁴ or y = x⁴ − 3x² + 2 therefore show a clear mirror symmetry.
当四次函数只含偶次项时，它是偶函数。此时一般形式变为：
f(x) = ax⁴ + cx² + e
偶四次函数关于 y 轴对称，即对任意 x 有 f(−x) = f(x)。例如 y = x⁴ 或 y = x⁴ − 3x² + 2 的图像显然具有镜像对称性。
If the polynomial contains any odd powers, the graph is not symmetric about the y-axis. In rare cases, a quartic can be point-symmetric about a stationary point, but this is not part of the standard A-Level syllabus.
若多项式含有任何奇次项，则图像不关于 y 轴对称。极少数情况下，四次函数可能关于某个驻点中心对称，但这不属于A-Level标准考纲。
5. First Derivative and Stationary Points | 一阶导数与驻点
The derivative of f(x) = ax⁴ + bx³ + cx² + dx + e is:
f′(x) = 4ax³ + 3bx² + 2cx + d
y points. Since f′(x) is a cubic, it has up to three real solutions. Therefore a quartic can have at most three turning points.
函数 f(x) = ax⁴ + bx³ + cx² + dx + e 的导数为：
f′(x) = 4ax³ + 3bx² + 2cx + d
令 f′(x) = 0 可得驻点。由于 f′(x) 是三次函数，它至多有三个实数解，因此四次函数至多有三个转向点。
A change of sign of f′(x) around a stationary point indicates a local maximum or minimum.
若 f′(x) 在驻点两侧变号，则该点为局部极大值或极小值。
If f′(x) does not change sign, the stationary point is a horizontal point of inflection.
若 f′(x) 不变号，则该驻点为水平拐点。
Because a quartic's derivative is cubic, the tangent can be horizontal at one, two, or three distinct points, producing different overall shapes.
因为四次函数的导数是三次函数，其水平切线可以在一处、两处或三处出现，从而产生不同的整体形状。
6. Second Derivative, Concavity and Points of Inflection | 二阶导数、凹凸性与拐点
The second derivative of a quartic is:
f″(x) = 12ax² + 6bx + 2c
ssible points of inflection. Since a quadratic has at most two real roots, a quartic can have at most two points of inflection.
四次函数的二阶导数为：
f″(x) = 12ax² + 6bx + 2c
它是二次函数。解 f″(x) = 0 可得到可能的拐点 x 坐标。二次函数至多有两个实根，因此四次函数至多有两个拐点。
If f″(x) > 0 on an interval, the graph is convex (curving upward).
若在某个区间内 f″(x) > 0，则图像是凸的（向上弯曲）。
If f″(x) < 0 on an interval, the graph is concave (curving downward).
若在某个区间内 f″(x) < 0，则图像是凹的（向下弯曲）。
At a point of inflection, f″(x) = 0 and the sign of f″(x) must change.
在拐点处，f″(x) = 0，且 f″(x) 的符号必须改变。
2x². This is zero at x = 0 but does not change sign, so (0,0) is not a point of inflection — it is a local minimum.
例如，f(x) = x⁴ 的二阶导数为 f″(x) = 12x²。在 x = 0 处为零，但符号不改变，因此 (0,0) 不是拐点，而是局部极小值点。
7. Systematic Steps for Sketching | 绘制图像的系统步骤
To sketch a quartic function accurately, follow these steps:
要精确绘制四次函数图像，请遵循以下步骤：
Determine the sign of a and the end behaviour.
确定 a 的符号及端部行为。
Find the y-intercept: set x = 0, giving y = e.
求 y 截距：令 x = 0，得 y = e。
Find the x-intercepts: solve f(x) = 0, factorising where possible.
求 x 截距：解 f(x) = 0，尽量因式分解。
e f′(x) = 0 to find stationary points; determine their nature using f″(x) or a sign table.
用 f′(x) = 0 求驻点，并用 f″(x) 或符号表判断其类型。
Find any points of inflection from f″(x) = 0.
由 f″(x) = 0 求出拐点。
Plot the identified points and connect them with a smooth curve, respecting symmetry and concavity.
标出上述点并用光滑曲线连接，注意对称性与凹凸性。
Example: Sketch f(x) = x⁴ − 5x² + 4
a = 1 > 0, so both ends go upward.
a = 1 > 0，所以两端向上。
y-intercept: f(0) = 4.
y 截距：f(0) = 4。
Factorise: x⁴ − 5x² + 4 = (x² − 1)(x² − 4) = (x−1)(x+1)(x−2)(x+2). Roots: ±1, ±2.
因式分解：x⁴ − 5x² + 4 = (x² − 1)(x² − 4) = (x−1)(x+1)(x−2)(x+2)。零点为 ±1、±2。
Stationary points: f′(x) = 4x³ − 10x = 2x(2x² − 5). Hence x = 0 and x = ±√5/2. Values: f(0) = 4, f(±√5/2) = −9/4. Local max at (0,4); local minima at (±√5/2, −9/4).
驻点：f′(x) = 4x³ − 10x = 2x(2x² − 5)。因此 x = 0 和 x = ±√5/2。对应函数值：f(0) = 4，f(±√5/2) = −9/4。局部极大值在 (0,4)，局部极小值在 (±√5/2, −9/4)。
8. Common Transformations | 常见变换
Quartic graphs can be transformed using standard function transformations.
四次函数图像可通过标准函数变换进行平移、伸缩和翻转。
Vertical translation: y = f(x) + k moves the graph up (k > 0) or down (k < 0).
垂直平移：y = f(x) + k 使图像向上（k > 0）或向下（k < 0）移动。
Horizontal translation: y = f(x − h) shifts the graph right by h units.
水平平移：y = f(x − h) 使图像向右移动 h 个单位。
Vertical stretch: y = m·f(x) stretches the graph vertically by factor m.
垂直伸缩：y = m·f(x) 使图像在垂直方向拉伸 m 倍。
Reflection in the x-axis: y = −f(x) flips the graph upside down.
关于 x 轴反射：y = −f(x) 使图像上下翻转。
y-axis: y = f(−x) flips the graph left-right. For an even quartic, the graph is unchanged.
关于 y 轴反射：y = f(−x) 使图像左右翻转。对于偶四次函数，图像不变。
− 2 is the graph of y = x⁴ shifted 1 unit left and 2 units down, with minimum at (−1, −2).
例如，y = (x + 1)⁴ − 2 是 y = x⁴ 的图像向左平移 1 个单位、向下平移 2 个单位得到的，最小值为 (−1, −2)。
9. Typical Exam Questions and Pitfalls | 典型题型与常见陷阱
In A-Level exams, quartic questions often ask you to solve equations, sketch graphs, or use the discriminant within a quartic context.
在A-Level考试中，四次函数题目通常要求解方程、绘制图像，或在四次函数情境中运用判别式。
he discriminant of this quadratic tells you how many real values of u exist, but each positive u gives two x-values.
换元技巧：将四次式看作关于 x² 的二次式。例如令 u = x²，则 x⁴ + px² + q = 0 变为 u² + pu + q = 0。该二次式的判别式决定 u 的实值个数，而每个正 u 对应两个 x 值。
Common pitfall: Forgetting that a quartic with a > 0 and no real roots lies entirely above the x-axis, so f(x) > 0 for all x.
常见陷阱：当 a > 0 且无实根时，图像完全位于 x 轴上方，即对所有 x 有 f(x) > 0。
Common pitfall: Confusing points of inflection with stationary points. A point of inflection does not require a horizontal tangent for a quartic.
常见陷阱：将拐点与驻点混淆。四次函数的拐点不要求切线水平。
Common pitfall: Drawing more than three turning points or more than two points of inflection. Remember the limits imposed by the derivative degrees.
常见陷阱：画出超过三个转向点或超过两个拐点。务必记住由导数次数带来的限制。
10. Summary | 总结
To master quartic graphs, keep these key facts in mind:
掌握四次函数图像，请牢记以下要点：
| Property | Result |
| Degree | 4 |
| Number of real roots | 0 to 4 |
| Stationary points | At most 3 |
| Points of inflection | At most 2 |
| End behaviour | Both ends same direction, sign of a |
| Symmetry | Only if no odd powers |
Practice by sketching a variety of quartics, using factorisation, derivatives and symmetry checks. With a clear method, you will quickly identify the essential shape of any quartic graph.
通过练习绘制多种四次函数图像，熟练运用因式分解、导数和对称性判断，你就能快速把握任意四次函数图像的核心形状。
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