Quadratic Functions Key Points | 二次函数考点精讲

📚 Quadratic Functions Key Points | 二次函数考点精讲

Quadratic functions form the backbone of many IB and Edexcel mathematics topics. They appear not only in algebra but also in calculus, coordinate geometry, and real‑world modelling. Mastering the standard form, vertex, discriminant, and the link between algebraic expressions and graphs will give you confidence across both papers. This revision guide walks you through every essential point.

二次函数是 IB 和 Edexcel 数学体系中极为重要的基础模块。无论代数、微积分还是坐标几何与实际问题建模，都离不开它。熟练掌握标准式、顶点、判别式以及代数表达式与图像的对应关系，能帮助你在考场上从容应对两类试卷。本文带你逐条梳理常考要点。

1. Definition and Standard Form | 定义与标准形式

A quadratic function is any function that can be written as f(x) = ax² + bx + c, where a, b and c are real constants and a ≠ 0. The graph is a parabola. If a > 0 the parabola opens upwards (u‑shaped), and if a < 0 it opens downwards (n‑shaped).

二次函数是指可写成 f(x) = ax² + bx + c 的函数，其中 a、b、c 为实数且 a ≠ 0。其图像是抛物线。当 a > 0 时抛物线开口向上，呈 “U” 形；当 a < 0 时开口向下，呈 “∩” 形。

The constant c gives the y‑intercept since f(0) = c. The leading coefficient a controls the “width” of the parabola — larger |a| makes the graph steeper, smaller |a| makes it flatter.

常数 c 是 y 轴截距，因为 f(0) = c。首项系数 a 还控制抛物线的 “宽窄”：|a| 越大图像越陡，|a| 越小图像越平坦。

2. Vertex Form and Completing the Square | 顶点式与配方法

Any quadratic can be rewritten in vertex form: f(x) = a(x − h)² + k, where (h, k) is the vertex. To convert from standard form, use completing the square. For f(x) = ax² + bx + c, factor a from the x‑terms, then add and subtract (b/(2a))² inside the bracket.

任何二次函数都可以改写为顶点式：f(x) = a(x − h)² + k，其中 (h, k) 即为顶点。将标准式改写时需要用配方法：对于 f(x) = ax² + bx + c，先把 a 提出，然后在括号内加上并减去 (b/(2a))²。

Example: f(x) = 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3. Vertex is (−2, −3).

例如：f(x) = 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3，顶点为 (−2, −3)。

Vertex form is ideal for identifying the maximum or minimum value. For a > 0, k is the minimum value; for a < 0, k is the maximum.

顶点式极适合用来确定最大或最小值。当 a > 0 时，k 是最小值；当 a < 0 时，k 是最大值。

In IB exams you are often asked to write a quadratic in the form a(x − p)² + q and hence state the coordinates of the vertex or the range of the function.

在 IB 考试中，经常要求将二次函数改写成 a(x − p)² + q 的形式，并由此写出顶点坐标或函数的值域。

3. Factored Form and Roots | 交点式与根

If a quadratic has real roots r₁ and r₂, it can be written as f(x) = a(x − r₁)(x − r₂). The roots are the x‑intercepts of the graph. To find roots, you can factorise, use the quadratic formula, or apply the zero product property.

若二次函数有实根 r₁ 与 r₂，则可写成 f(x) = a(x − r₁)(x − r₂)，这两个根就是图像与 x 轴的交点。求根时可因式分解、使用求根公式或利用零乘积性质。

The quadratic formula: x = [−b ± √(b² − 4ac)] / (2a). It always works, even when the quadratic does not factorise nicely.

求根公式：x = [−b ± √(b² − 4ac)] / (2a)。它永远有效，即便二次式不能漂亮地分解。

When the discriminant b² − 4ac is negative, there are no real roots, so the factored form over real numbers does not exist. In that case the graph never touches the x‑axis.

当判别式 b² − 4ac 为负时，没有实根，因此在实数范围内不存在交点式，图像与 x 轴无交点。

4. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant Δ = b² − 4ac determines the number and type of real roots:

Δ > 0: two distinct real roots (graph cuts the x‑axis twice).
Δ = 0: exactly one real root (a repeated root; the graph touches the x‑axis at the vertex).
Δ < 0: no real roots (graph lies entirely above or below the x‑axis).

判别式 Δ = b² − 4ac 决定了实根的个数与性质：

Δ > 0：两个不等实根（图像与 x 轴交于两点）。
Δ = 0：一个实根（重根；图像在顶点处与 x 轴相切）。
Δ < 0：无实根（图像完全在 x 轴上方或下方）。

Edexcel problems often ask you to find the set of values of a parameter (often k) such that the quadratic has two distinct roots, a repeated root, or no real roots. Set the discriminant condition accordingly.

Edexcel 试题常要求确定参数（通常为 k）的取值范围，使得二次函数有两个相异实根、一个重根或无实根。根据题意列出相应的判别式不等式即可。

Also remember that for tangency problems (line touches parabola), set the discriminant of the combined equation to zero.

另外要记住，在切线问题中（直线与抛物线相切），将联立方程的判别式设为零。

5. Axis of Symmetry and Vertex Coordinates | 对称轴与顶点坐标

Every parabola has an axis of symmetry, a vertical line passing through the vertex. For f(x) = ax² + bx + c, the axis of symmetry is x = −b/(2a). Plugging this x‑value into the function gives the vertex ( −b/(2a), f(−b/(2a)) ).

每条抛物线都有一条通过顶点的铅垂对称轴。对于 f(x) = ax² + bx + c，对称轴为 x = −b/(2a)。将该 x 值代入函数即得顶点 ( −b/(2a), f(−b/(2a)) )。

In vertex form f(x) = a(x − h)² + k, the axis is x = h and the vertex is (h, k). Many questions ask you to find the line of symmetry and the vertex directly from a given form.

在顶点式 f(x) = a(x − h)² + k 中，对称轴为 x = h，顶点为 (h, k)。许多题目会要求你从给定形式直接写出对称轴与顶点。

Understanding symmetry helps in solving quadratic equations graphically and in finding midpoint coordinates of the x‑intercepts: the x‑coordinate of the vertex is the average of the roots.

理解对称性有助于图像法解方程，也有助于求 x 轴交点中点：顶点的 x 坐标正是两个根的平均值。

6. Graphing Quadratic Functions | 二次函数图像

To sketch a quadratic, determine: (1) the direction of opening (sign of a), (2) the y‑intercept (c), (3) the x‑intercepts (roots, if real), and (4) the vertex. Then draw a smooth curve through these key points. Label all intercepts and the vertex clearly.

绘制二次函数图像时，需确定：(1) 开口方向（a 的正负），(2) y 轴截距 (c)，(3) x 轴截距（实根，如果有的话），(4) 顶点。然后通过这几个关键点描出平滑曲线，并清楚地标出截距与顶点。

In Edexcel examinations sketch questions often carry marks for correct shape, correct position relative to axes, and correct labelling. Do not worry about exact scaling but maintain symmetry.

在 Edexcel 考试中，画图题往往依据形状是否正确、相对坐标轴的位置是否正确以及是否标注来给分。不必精确按比例，但须保持对称。

For parabolas with a repeated root, the vertex lies on the x‑axis. For ones with no real roots, the graph floats above or below the x‑axis with the vertex being the highest or lowest point.

对于有重根的抛物线，顶点落在 x 轴上；没有实根时，图像悬在 x 轴上方或下方，顶点是最高或最低点。

7. Transformations of Quadratic Graphs | 二次函数图像的变换

Transformations include translations: f(x) → f(x) + d shifts the graph vertically by d; f(x) → f(x + d) shifts horizontally by −d. Stretches: f(x) → p f(x) stretches vertically by factor p; f(x) → f(q x) stretches horizontally by factor 1/|q|. Reflections: f(x) → −f(x) reflects in the x‑axis; f(x) → f(−x) reflects in the y‑axis.

图像变换包含平移：f(x) → f(x) + d 使图像竖直移动 d；f(x) → f(x + d) 使图像水平移动 −d。伸缩：f(x) → p f(x) 竖直方向伸缩 p 倍；f(x) → f(q x) 水平方向伸缩 1/|q| 倍。反射：f(x) → −f(x) 关于 x 轴对称；f(x) → f(−x) 关于 y 轴对称。

In vertex form f(x) = a(x − h)² + k, you can directly read the sequence of transformations from the parent function y = x²: horizontal translation h, vertical translation k, vertical stretch |a|, and reflection if a < 0.

对于顶点式 f(x) = a(x − h)² + k，可以直接从母函数 y = x² 看出变换顺序：水平平移 h，竖直平移 k，竖直伸缩 |a|，以及若 a < 0 则关于 x 轴反射。

Edexcel frequently tests the combination of transformations and asks you to write the equation of a transformed quadratic. Always apply horizontal shifts before stretches or reflections in combined transformations (but follow standard function transformation rules).

Edexcel 常考组合变换，并会要求写出变换后的二次方程。在进行组合变换时，通常先进行水平平移，再进行伸缩或反射（但需遵循函数变换的通则）。

8. Quadratic Inequalities | 二次不等式

Solving a quadratic inequality such as ax² + bx + c > 0 involves first finding the real roots (if any) of the corresponding equation. The roots divide the number line into intervals. Test a value from each interval to determine where the inequality holds. Alternatively, sketch a quick graph: the quadratic is positive where the graph is above the x‑axis, negative where below.

解二次不等式如 ax² + bx + c > 0 时，先求出对应二次方程的实根（如果有的话）。这些根把数轴分成若干区间，在每个区间内取一个测试值判断不等式是否成立。或者快速画出草图：图像在 x 轴上方时函数为正，下方时为负。

For a > 0, the parabola opens upwards, so the expression is positive outside the roots and negative between the roots. For a < 0, the pattern is reversed. Remember to use open or closed intervals depending on strict or non‑strict inequality.

当 a > 0 时抛物线开口向上，函数值在两根之外为正，两根之间为负；a < 0 时情形相反。注意根据是否包含等号使用开区间或闭区间。

Common exam tasks: find the set of x for which a given quadratic is always positive, or determine parameter values that make a quadratic inequality hold for all real x. In the latter case, require a > 0 and Δ < 0.

常见考试任务：求使给定二次式恒为正的 x 集合，或确定使二次不等式对一切实数 x 成立的参数取值范围。后者需满足 a > 0 且 Δ < 0。

9. Applications and Word Problems | 应用与文字题

Quadratic functions model projectile motion, area optimisation, profit/cost analysis, and geometric properties. Typically you will be given a scenario, asked to write a quadratic model, and then find the maximum/minimum value or a specific coordinate. Use vertex form or −b/(2a) for optimisation.

二次函数常用来建模抛体运动、面积最优化、利润/成本分析以及几何性质。通常会给出情景，要求你写出二次模型，进而求最大值/最小值或特定坐标。优化时使用顶点式或 −b/(2a)。

For example, the height of a ball thrown upwards is often given by h(t) = −4.9t² + vt + h₀. The time to maximum height is −v/(2 × (−4.9)) = v/9.8, and the maximum height is h(t) at that t.

例如，上抛小球的飞行高度常表示为 h(t) = −4.9t² + vt + h₀。达到最高点的时间为 −v/(2 × (−4.9)) = v/9.8，最高点为该时刻的 h(t) 值。

In area problems, you may express length and width in terms of the same variable, multiply to get a quadratic area function, then find its vertex. Always check that the variable’s physical meaning restricts the domain.

在面积问题中，可将长和宽用同一个变量表示，相乘得到二次面积函数，再求其顶点。务必根据实际意义检查变量的定义域限制。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Mistake 1: Forgetting the negative sign when applying −b/(2a). Double‑check the sign of b.

常见错误一：应用 −b/(2a) 时漏掉负号。务必检查 b 的符号。

Mistake 2: Confusing the sign of c with the y‑intercept — the intercept is (0, c), always.

常见错误二：混淆 c 的符号与 y 截距——截距总是 (0, c)。

Mistake 3: Writing the vertex from vertex form as (h, k) when the form is a(x − h)² + k — note the minus sign inside the square.

常见错误三：从顶点式 a(x − h)² + k 写顶点时误为 (h, k)——注意括号内是减号。

Mistake 4: Incorrectly expanding or factorising, especially missing the a factor. Always expand back to verify.

常见错误四：展开或因式分解时漏掉系数 a，尤其是在配方后。可通过回代展开验证。

Mistake 5: When solving inequalities, dividing by a negative a requires reversing the inequality sign — be careful when moving terms.

常见错误五：解不等式时，除以负因数 a 会导致不等号方向反转——移项时要小心。

Top tip: In graphing questions, always plot the vertex first, then the y‑intercept, and use symmetry to find a matching point on the other side. This gives accurate sketches even without precise roots.

重要技巧：画图时先标出顶点，再标 y 截距，然后利用对称性找到另一侧的对应点。即使没有精确根值也能画出准确草图。

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Quadratic Functions Key Points | 二次函数考点精讲