Quadratic Functions and Graphs | 二次函数与图像

📚 Quadratic Functions and Graphs | 二次函数与图像

Quadratic functions are among the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, and problem-solving questions. Mastering the standard form, factorisation, completing the square, and graph sketching will give you a strong foundation for both Paper 2 and Paper 4.

二次函数是 IGCSE 数学大纲中最重要的内容之一。它出现在代数、坐标几何和应用题中。掌握标准形式、因式分解、配方法以及图像绘制，将为你应对 Paper 2 和 Paper 4 打下坚实基础。

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1. The General Form of a Quadratic Function | 二次函数的一般形式

A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a, b and c are constants and a ≠ 0. The value of a determines whether the parabola opens upward or downward.

二次函数是指可以写成 f(x) = ax² + bx + c 形式的函数，其中 a、b、c 为常数，且 a ≠ 0。a 的值决定了抛物线的开口方向。

f(x) = ax² + bx + c, a ≠ 0

• If a > 0, the parabola opens upward and has a minimum point.

  如果 a > 0，抛物线开口向上，有最小值点。

• If a < 0, the parabola opens downward and has a maximum point.

  如果 a < 0，抛物线开口向下，有最大值点。

• The curve is always symmetric about a vertical line called the axis of symmetry.

  曲线始终关于一条称为对称轴的竖直线对称。

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2. Roots and the Discriminant | 根与判别式

The roots of a quadratic equation are the x-values where the graph crosses the x-axis. They can be found by factorising, using the quadratic formula, or completing the square.

二次方程的根是图像与 x 轴交点处的 x 值。可以通过因式分解、求根公式或配方法来求解。

x = (−b ± √(b² − 4ac)) / 2a

The expression b² − 4ac is called the discriminant. It tells us how many real roots the equation has:

表达式 b² − 4ac 称为判别式。它告诉我们方程有多少个实数根：

Discriminant b² − 4ac	Number of Real Roots	Graph Intersection with x-axis
Positive ( > 0 )	Two distinct roots	Two intersection points
Zero ( = 0 )	One repeated root	One touching point
Negative ( < 0 )	No real roots	No intersection

• If discriminant > 0: two distinct real roots.

  若判别式 > 0：两个不相等的实数根。

• If discriminant = 0: one repeated root (graph touches the axis).

  若判别式 = 0：一个重根（图像与轴相切）。

• If discriminant < 0: no real roots.

  若判别式 < 0：无实数根。

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3. The Vertex (Turning Point) | 顶点（极值点）

Every parabola has exactly one vertex. This is the point where the curve changes direction. If the parabola opens upward, the vertex is the minimum point; if it opens downward, the vertex is the maximum point.

每条抛物线都只有一个顶点。这是曲线改变方向的点。如果抛物线开口向上，顶点为最小值点；如果开口向下，顶点为最大值点。

For f(x) = ax² + bx + c, the x-coordinate of the vertex is given by:

对于 f(x) = ax² + bx + c，顶点的 x 坐标为：

x = −b / 2a

Once you have this x-value, substitute it back into the original function to find the corresponding y-coordinate. The vertex is then written as (x, y).

求出 x 值后，将其代入原函数即可得到对应的 y 坐标。顶点坐标即为 (x, y)。

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4. Axis of Symmetry | 对称轴

The axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two mirror-image halves.

对称轴是经过顶点的一条竖直线。它将抛物线分成互为镜像的两半。

x = −b / 2a

Notice that the axis of symmetry has the same equation as the x-coordinate of the vertex. This makes sense because the vertex always lies on the axis of symmetry.

注意，对称轴的方程与顶点 x 坐标的公式完全相同。这很合理，因为顶点始终位于对称轴上。

• The axis of symmetry is always vertical.

  对称轴始终是竖直线。

• Every point on the parabola has a mirror point on the opposite side of this axis.

  抛物线上的每一个点在对称轴的另一侧都有一个镜像点。

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5. Completing the Square | 配方法

Completing the square is a powerful technique for rewriting a quadratic function in vertex form:

配方法是一种强大的技巧，可以将二次函数改写为顶点形式：

f(x) = a(x − h)² + k

In this form, the vertex is simply (h, k). The value of h is the x-coordinate of the vertex, and k is the y-coordinate (the minimum or maximum value of the function).

在这种形式下，顶点坐标就是 (h, k)。其中 h 是顶点的 x 坐标，k 是 y 坐标（函数的最小值或最大值）。

For example, to complete the square for f(x) = x² + 6x + 5:

例如，对 f(x) = x² + 6x + 5 配方：

x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4

Therefore, the vertex is at (−3, −4), and the axis of symmetry is x = −3.

因此，顶点坐标为 (−3, −4)，对称轴为 x = −3。

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6. Sketching a Quadratic Graph | 绘制二次函数图像

To sketch a quadratic graph accurately, you need to identify four key features:

要准确地绘制二次函数图像，你需要确定四个关键特征：

Feature	How to Find It
y-intercept	Set x = 0, read c from f(x) = ax² + bx + c
x-intercepts (roots)	Set f(x) = 0 and solve
Vertex	Use x = −b/2a, then substitute
Direction of opening	Look at the sign of a

• Plot the y-intercept first, then the x-intercepts if they exist, then the vertex.

  先标出 y 轴截距，然后是 x 轴截距（如果存在），最后标出顶点。

• Draw a smooth U-shaped or ∩-shaped curve through these points.

  用平滑的 U 形或 ∩ 形曲线连接这些点。

• Always label the vertex, axis of symmetry, and intercepts clearly.

  始终清晰地标注顶点、对称轴和截距。

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7. Solving Quadratic Equations by Factorisation | 用因式分解解二次方程

Factorisation is the fastest method when the quadratic expression can be factorised. For a quadratic in the form x² + bx + c, look for two numbers that multiply to give c and add to give b.

当二次表达式可以因式分解时，因式分解是最快的方法。对于 x² + bx + c 形式的二次式，找两个数，它们相乘等于 c，相加等于 b。

For example, solve x² − 5x + 6 = 0:

例如，解 x² − 5x + 6 = 0：

(x − 2)(x − 3) = 0

Therefore, x = 2 or x = 3.

因此，x = 2 或 x = 3。

• Always factor out any common factor first before attempting to factorise the quadratic.

  在尝试因式分解二次式之前，先提取公因式。

• If the coefficient of x² is not 1, use the method of splitting the middle term or use the quadratic formula.

  如果 x² 的系数不为 1，使用拆中项法或求根公式。

• Check your answers by substituting back into the original equation.

  通过代回原方程来检查答案。

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8. The Quadratic Formula | 求根公式

When a quadratic equation cannot be factorised easily, use the quadratic formula. This formula works for all quadratic equations, including those with irrational roots.

当二次方程不易因式分解时，使用求根公式。这个公式适用于所有二次方程，包括具有无理数根的情况。

x = (−b ± √(b² − 4ac)) / 2a

For example, solve 2x² + 3x − 1 = 0:

例如，解 2x² + 3x − 1 = 0：

Here a = 2, b = 3, c = −1. Substituting into the formula gives:

这里 a = 2，b = 3，c = −1。代入公式得：

x = (−3 ± √(9 − 4 × 2 × (−1))) / 4 = (−3 ± √17) / 4

So x ≈ 0.281 or x ≈ −1.781 (to 3 decimal places).

因此 x ≈ 0.281 或 x ≈ −1.781（精确到三位小数）。

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9. Applications in Real-World Problems | 实际应用问题

Quadratic functions model many real-world situations, such as projectile motion, area optimisation, and profit calculations. In IGCSE exams, these appear as word problems where you must form a quadratic equation and solve it.

二次函数可以建模许多实际情况，如抛体运动、面积优化和利润计算。在 IGCSE 考试中，这些问题以应用题的形式出现，你需要建立二次方程并求解。

For example, the height h (in metres) of a ball after t seconds is given by h = −5t² + 20t + 1. To find when the ball hits the ground, solve −5t² + 20t + 1 = 0.

例如，球在 t 秒后的高度 h（米）由 h = −5t² + 20t + 1 给出。要求球落地的时间，即解 −5t² + 20t + 1 = 0。

• Read the problem carefully and identify what variable is x and what expression is f(x).

  仔细阅读题目，确定哪个变量是 x，哪个表达式是 f(x)。

• Decide whether a negative root makes sense in the context of the problem.

  判断负根在题目背景下是否有意义。

• Always write the final answer with the correct units.

  最终答案始终要写上正确的单位。

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10. Graphs of y = (x − p)² + q and Transformations | y = (x − p)² + q 的图像与变换

The vertex form directly reveals transformations of the basic graph y = x². The value of p shifts the graph horizontally, and q shifts it vertically.

顶点形式直接揭示了基本图像 y = x² 的变换。p 的值使图像水平移动，q 使图像垂直移动。

Transformation	Equation	Effect
Horizontal shift right by p	y = (x − p)²	Vertex moves to (p, 0)
Horizontal shift left by p	y = (x + p)²	Vertex moves to (−p, 0)
Vertical shift up by q	y = x² + q	Vertex moves to (0, q)
Vertical shift down by q	y = x² − q	Vertex moves to (0, −q)

• A negative sign before the square term, y = −x², reflects the graph in the x-axis.

  平方项前有负号，即 y = −x²，表示图像关于 x 轴翻转。

• Combining transformations: y = (x − 3)² + 2 means shift right by 3 and up by 2.

  组合变换：y = (x − 3)² + 2 表示向右平移 3 个单位，再向上平移 2 个单位。

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11. Common Exam Pitfalls | 常见考试误区

Students often lose marks in quadratic function questions due to a few repeated mistakes. Being aware of these pitfalls can help you avoid them.

学生在二次函数题目中常因几个反复出现的错误而失分。意识到这些误区可以帮助你避免它们。

• Forgetting that a ≠ 0 in a quadratic function.

  忘记二次函数中 a ≠ 0 的条件。

• Mixing up the sign of h in the vertex form y = a(x − h)² + k.

  混淆顶点形式 y = a(x − h)² + k 中 h 的符号。

• When factorising, forgetting to set each factor to zero separately.

  因式分解后，忘记分别令每个因式为 0。

• Substituting incorrectly into the quadratic formula, especially the sign of c.

  代入求根公式时出错，尤其是 c 的符号。

• Sketching the graph without finding the vertex.

  绘制图像时没有求出顶点。

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12. Exam Preparation Checklist | 考前复习清单

Use this checklist to confirm you are fully prepared for quadratic function questions in your IGCSE exam.

使用这份清单来确认你已经为 IGCSE 考试中的二次函数题目做好充分准备。

• I can identify the values of a, b and c from a quadratic expression.

  我能从二次表达式中识别 a、b 和 c 的值。

• I can calculate the discriminant and state the number of roots.

  我能计算判别式并判断根的个数。

• I can find the vertex and axis of symmetry.

  我能求出顶点坐标和对称轴。

• I can complete the square and rewrite in vertex form.

  我能用配方法将函数改写为顶点形式。

• I can sketch a quadratic graph with all key features labelled.

  我能绘制二次函数图像并标出所有关键特征。

• I can solve quadratic equations by factorisation, formula and completing the square.

  我能用因式分解、求根公式和配方法解二次方程。

• I understand how transformations affect the graph of y = x².

  我理解变换对 y = x² 图像的影响。

• I can apply quadratic equations to solve word problems.

  我能应用二次方程解决应用题。

Practice makes permanent. Make sure you attempt a wide variety of questions, including past paper questions, before your exam day.

熟能生巧。确保在考试前练习各类题目，包括历年真题。

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