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-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)

Each factor corresponds to an x-intercept. Repeated roots produce tangency to the x-axis. For example, (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

An even quartic is symmetric about the y-axis. This means f(−x) = f(x) for all x. Graphs 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

Setting f′(x) = 0 gives the stationary 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

It is a quadratic function. Solving f″(x) = 0 gives the x-coordinates of possible 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。

For example, f(x) = x⁴ has f″(x) = 12x². 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。
Use 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。
Reflection in the 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。

For example, y = (x + 1)⁴ − 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。

Discriminant trick: Write a quartic as a quadratic in x². For example, u = x² turns x⁴ + px² + q = 0 into u² + pu + q = 0. The 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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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