📚 IGCSE Maths G-7 Teacher Guide: Quadratic Functions and Graphs | IGCSE 数学 G-7 教师用书：二次函数与图像
This teacher guide supports IGCSE Mathematics planning for the topic of quadratic functions and graphs. It explains key teaching points, common difficulties, and assessment strategies linked to both the core and extended curriculum.
本教师用书支持 IGCSE 数学中二次函数与图像的教学规划。它解释关键教学点、常见难点以及与核心和拓展课程相关的评估策略。
1. Learning Objectives | 学习目标
By the end of this topic, students should be able to recognise a quadratic expression, write it in standard form, and identify the coefficient conditions for a quadratic.
学完本专题后，学生应能识别二次表达式，将其写成标准形式，并确定二次式的系数条件。
They should also be able to solve quadratic equations by factorising, completing the square, and using the quadratic formula, and interpret the discriminant correctly.
他们还应能通过因式分解、配方法和求根公式解二次方程，并正确解释判别式。
atic graphs, including the vertex, axis of symmetry, roots, and transformations of y = x².
最后，学生需要绘制并分析二次函数图像，包括顶点、对称轴、根以及 y = x² 的图像变换。
2. Quadratic Expressions and Standard Form | 二次表达式与标准式
A quadratic expression in x is an expression of degree 2. Its standard form is written as ax² + bx + c, where a, b, and c are constants and a ≠ 0.
关于 x 的二次表达式是次数为 2 的表达式。其标准形式写作 ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。
ion with a = 3, b = −5, and c = 2. If a = 0, the expression becomes linear, not quadratic.
例如，3x² − 5x + 2 是一个二次表达式，其中 a = 3，b = −5，c = 2。如果 a = 0，该表达式就变成一次式，而不是二次式。
Teachers should emphasise that the term 'quadratic' comes from the square term x², and the coefficient a must be non-zero.
教师应强调“二次”一词来源于平方项 x²，且系数 a 必须不为零。
y = ax² + bx + c, a ≠ 0
3. Completing the Square | 配方法
Completing the square rewrites a quadratic expression x² + bx + c in the form (x + p)² + q. This method reveals the vertex of the graph and helps solve equations.
配方法将二次表达式 x² + bx + c 改写为 (x + p)² + q 的形式。此方法能揭示图像的顶点，并有助于解方程。
identity is x² + bx = (x + b/2)² − (b/2)². For example, x² + 6x + 5 becomes (x + 3)² − 4.
关键恒等式是 x² + bx = (x + b/2)² − (b/2)²。例如，x² + 6x + 5 可化为 (x + 3)² − 4。
terms. For example, 3x² + 12x + 7 = 3(x² + 4x) + 7 = 3[(x + 2)² − 4] + 7 = 3(x + 2)² − 5.
当 x² 的系数不是 1 时，首先将其从 x² 项和 x 项中提出。例如，3x² + 12x + 7 = 3(x² + 4x) + 7 = 3[(x + 2)² − 4] + 7 = 3(x + 2)² − 5。
Common mistakes include sign errors when subtracting the square and forgetting to multiply the constant by the factorised coefficient.
常见错误包括减去平方项时的符号错误，以及忘记将常数乘以提出来的系数。
4. Quadratic Formula and Discriminant | 求根公式与判别式
ic equation ax² + bx + c = 0 with a ≠ 0, the solutions are given by the quadratic formula.
对于任意二次方程 ax² + bx + c = 0 且 a ≠ 0，其解由求根公式给出。
x = (−b ± √b² − 4ac) ÷ (2a)
are root, Δ = b² − 4ac, is called the discriminant. It determines the nature of the roots.
根号下的表达式 Δ = b² − 4ac 称为判别式。它决定根的性质。
- If Δ > 0, the equation has two distinct real roots.
- 如果 Δ > 0，方程有两个不同的实根。
- If Δ = 0, the equation has one repeated real root.
- 如果 Δ = 0，方程有一个重实根。
- If Δ < 0, the equation has no real roots.
- 如果 Δ < 0，方程没有实根。
Teachers should encourage students to calculate the discriminant before solving, as it gives quick insight into the number of solutions.
教师应鼓励学生在求解之前先计算判别式，因为它能快速判断解的个数。
5. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上且有最小值点；如果 a < 0，抛物线开口向下且有最大值点。
s on this line, and its y-coordinate is found by substituting this x-value into the function.
对称轴始终是竖直线 x = −b ÷ (2a)。顶点位于这条线上，其 y 坐标可通过将该 x 值代入函数求得。
or example, for y = x² − 4x + 3, the axis of symmetry is x = 2, and the vertex is (2, −1).
例如，对于 y = x² − 4x + 3，对称轴是 x = 2，顶点是 (2, −1)。
| Condition | Shape | Vertex |
| a > 0 | Opens upwards | Minimum point |
| a < 0 | Opens downwards | Maximum point |
6. Vertex and Axis of Symmetry | 顶点与对称轴
ion is written in completed square form y = a(x − h)² + k, the vertex is the point (h, k).
当二次函数写成配方式 y = a(x − h)² + k 时，顶点为点 (h, k)。
rtical line x = h. This form makes it easy to sketch the graph without expanding brackets.
对称轴是竖直线 x = h。这种形式使绘图变得容易，无需展开括号。
2(x − 3)² + 1 has vertex (3, 1), and the parabola opens upwards because a = 2 is positive.
例如，y = 2(x − 3)² + 1 的顶点是 (3, 1)，由于 a = 2 为正，抛物线开口向上。
onfuse the sign of h. The form is (x − h), so y = (x + 4)² has vertex (−4, 0), not (4, 0).
学生经常混淆 h 的符号。形式是 (x − h)，因此 y = (x + 4)² 的顶点是 (−4, 0)，而不是 (4, 0)。
7. Roots and x-Intercepts | 根与 x 轴交点
+ bx + c = 0 are the x-values where the graph crosses the x-axis. At these points, y = 0.
二次方程 ax² + bx + c = 0 的根是图像与 x 轴相交时的 x 值。在这些点处，y = 0。
0, then the roots are x = p and x = q. The graph crosses the x-axis at (p, 0) and (q, 0).
如果二次式可以因式分解为 (x − p)(x − q) = 0，则根为 x = p 和 x = q。图像在 (p, 0) 和 (q, 0) 处与 x 轴相交。
If the discriminant is zero, the graph touches the x-axis at one point, which is also the vertex. If the discriminant is negative, there are no x-intercepts.
如果判别式为零，图像在一点处与 x 轴相切，该点也是顶点。如果判别式为负，则没有 x 轴交点。
Teachers should connect algebraic solutions to graphical behaviour so students see roots as points of intersection rather than abstract numbers.
教师应将代数解与图像行为联系起来，让学生将根视为交点，而不是抽象的数字。
8. Transformations of y = x² | y = x² 的图像变换
ts transformations follow a consistent pattern that students can use to sketch related functions.
最基本的二次函数图像是 y = x²。其变换遵循一致的模式，学生可以利用该模式绘制相关函数。
ranslated h units horizontally and k units vertically. Positive h moves the graph right, positive k moves it up.
对于 y = (x − h)² + k，y = x² 的图像水平平移 h 个单位、竖直平移 k 个单位。h 为正时图像向右移，k 为正时图像向上移。
For y = ax², the value of a controls stretching and reflection. If |a| > 1, the graph is narrower than y = x². If 0 < |a| < 1, it is wider. If a < 0, the graph is reflected in the x-axis.
对于 y = ax²，a 的值控制拉伸和反射。若 |a| > 1，图像比 y = x² 更窄；若 0 < |a| < 1，图像更宽；若 a < 0，图像关于 x 轴反射。
kill is to describe the combined transformation from y = x² to y = a(x − h)² + k in words.
一个常见的可迁移技能是用语言描述从 y = x² 到 y = a(x − h)² + k 的组合变换。
9. Solving Quadratic Inequalities | 二次不等式
6 > 0, first solve the corresponding equation x² − 5x + 6 = 0 to find the critical values.
要解二次不等式如 x² − 5x + 6 > 0，首先解相应的方程 x² − 5x + 6 = 0，以找到临界值。
parabola opens upwards, the expression is positive outside the interval between the roots.
此处 x² − 5x + 6 = (x − 2)(x − 3)，所以根为 x = 2 和 x = 3。由于抛物线开口向上，表达式在两根之间的区间外取正值。
Therefore the solution is x < 2 or x > 3. If the inequality were x² − 5x + 6 < 0, the solution would be 2 < x < 3.
因此解为 x < 2 或 x > 3。如果不等式是 x² − 5x + 6 < 0，解则是 2 < x < 3。
Students should be reminded to sketch a quick sign diagram or graph, especially when the coefficient a is negative, as this reverses the regions.
应提醒学生快速绘制符号图或图像，尤其是当系数 a 为负时，因为这会调换取值区域。
10. Applications and Modelling | 应用与建模
Quadratic functions commonly model projectile motion, areas of rectangles, and profit or cost problems in real life.
二次函数通常用于模拟现实生活中的抛体运动、矩形面积以及利润或成本问题。
elled by h = −5t² + 20t + 1, where h is the height in metres and t is the time in seconds.
例如，向上抛出的小球高度可建模为 h = −5t² + 20t + 1，其中 h 是高度（米），t 是时间（秒）。
t = −b ÷ (2a) gives t = −20 ÷ (2 × −5) = 2 seconds, and h(2) = −5(4) + 40 + 1 = 21 metres.
最大高度在顶点处求得。使用 t = −b ÷ (2a) 得 t = −20 ÷ (2 × −5) = 2 秒，且 h(2) = −5(4) + 40 + 1 = 21 米。
equires the quadratic formula or factorisation. This connects roots to real-world meaning.
小球落地时间可通过解 h = 0 求得，这需要使用求根公式或因式分解。这样就将根与现实意义联系起来。
11. Common Misconceptions | 常见误区
One common misconception is that every quadratic equation can be solved by simple factorisation. Students should know when to use the quadratic formula or completing the square.
一个常见误区是认为每个二次方程都可以通过简单的因式分解求解。学生应知道何时使用求根公式或配方法。
Another common error is writing the vertex as ( −b/2a, b/2a ) instead of substituting the x-value back into the function. The y-coordinate requires evaluation.
另一个常见错误是将顶点写作 ( −b/2a, b/2a )，而没有将 x 值代回函数求 y 坐标。y 坐标需要通过函数计算。
Students also confuse the direction of translations, for example believing that (x − 3)² shifts the graph left. In fact it shifts the graph right by 3 units.
学生还会混淆平移方向，例如认为 (x − 3)² 使图像向左平移。实际上它使图像向右平移 3 个单位。
Finally, some learners forget to check the sign of a when solving quadratic inequalities, which can reverse the solution set.
最后，一些学习者在解二次不等式时忘记检查 a 的符号，这可能导致解集反转。
12. Assessment Tips for Teachers | 教师评估建议
Assess students with a mix of procedural exercises, graph-sketching tasks, and real-world modelling problems to test both skills and understanding.
教师可通过程序性练习、图像绘制任务和现实建模问题相结合的方式来评估学生，以同时考查技能和理解。
Use open questions such as 'How does the discriminant tell you about the graph?' to probe conceptual understanding rather than memory.
使用开放式问题，如“判别式如何告诉你图像的情况？”来考查概念理解，而不是单纯记忆。
Encourage students to check their answers using a graphing tool or by substituting roots back into the original equation.
鼓励学生使用绘图工具或将根代回原方程来检查答案。
When marking, give credit for correct method even if the final answer has a minor arithmetic error, as IGCSE mark schemes often reward method marks.
评分时，即使最终答案有小的算术错误，只要方法正确也应给分，因为 IGCSE 评分方案通常奖励方法分。
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