📚 Mastering Quadratic Functions & Graphs | 掌握二次函数与图像
Quadratic functions are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and even in problem-solving questions. Understanding their graphs, roots, and transformations is essential for achieving top marks.
二次函数是 IGCSE 数学中最重要的考点之一。它不仅出现在代数和坐标几何中，也常出现在综合应用题里。掌握图像、根以及图像变换，是拿到高分的关键。
1. The Standard Form of a Quadratic Function | 二次函数的标准形式
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 highest power of x is always 2.
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。变量 x 的最高次数始终为 2。
- If a > 0, the graph opens upward (a "happy" parabola).
- 若 a > 0，抛物线开口向上，形成 "U" 形。
- If a < 0, the graph opens downward (a "sad" parabola).
- 若 a < 0，抛物线开口向下，形成 "∩" 形。
- The value of c is the y-intercept of the graph.
- 常数 c 的值就是抛物线与 y 轴的交点纵坐标。
2. Plotting Quadratic Graphs | 绘制二次函数图像
To plot a quadratic graph, we create a table of values for x and compute the corresponding f(x) values. We then plot the points and join them with a smooth curve.
绘制二次函数图像时，我们先列出自变量 x 的函数值表，计算出对应的 f(x)，再在坐标系中描点，并用平滑曲线连接各点。
| x | -2 | -1 | 0 | 1 | 2 | 3 |
| y = x² - 2x - 3 | 5 | 0 | -3 | -4 | -3 | 0 |
) = 0, and both points lie on the x-axis. This symmetry is a key feature of all parabolas.
注意图像的对称性：f(-1) = f(3) = 0，这两个点都在 x 轴上。对称性是所有抛物线的重要特征。
3. Roots of a Quadratic Equation | 二次方程的根
x) = 0. A quadratic equation can have two real roots, one repeated root, or no real roots.
二次方程的根（或解）是函数图像与 x 轴交点的横坐标，即 f(x) = 0 时的 x 值。二次方程可以有两个不相等的实数根、一个重根，或者没有实数根。
f(x) = ax² + bx + c = 0
- Two distinct real roots: the graph crosses the x-axis at two points.
- 两个不同的实数根：图像与 x 轴相交于两点。
- One repeated root: the graph touches the x-axis at one point (the vertex).
- 一个重根：图像与 x 轴相切于一点（即顶点处）。
- No real roots: the graph does not touch the x-axis at all.
- 无实数根：图像完全不与 x 轴相交。
4. Factorisation Method | 因式分解法
One of the fastest ways to solve a quadratic equation is by factorisation. We rewrite the equation as a product of two linear factors and set each factor equal to zero.
解二次方程最快的方法之一就是因式分解法。我们将方程改写为两个一次因式的乘积，然后令每个因式分别等于零。
x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0
Therefore x = 2 or x = 3. Always expand your factors to check your work quickly.
因此 x = 2 或 x = 3。解完后将因式展开，可以快速验证结果是否正确。
5. The Quadratic Formula | 求根公式
When factorisation is difficult or impossible, we use the quadratic formula. This formula works for any quadratic equation with real coefficients.
当因式分解困难或无法进行时，我们可以使用求根公式。该公式适用于任何实系数二次方程。
x = (-b ± √b² - 4ac) / 2a
This formula is given on the IGCSE formula sheet, but you must know how to substitute and simplify correctly, including dealing with surds.
这个公式会在 IGCSE 公式表中给出，但你必须掌握正确的代入与化简方法，包括如何处理根号中的无理数。
6. The Discriminant Δ | 判别式 Δ
The expression b² - 4ac inside the square root is called the discriminant. It tells us about the nature of the roots without solving the equation.
求根公式中根号内的表达式 b² - 4ac 称为判别式。它可以帮助我们在不解方程的情况下判断根的性质。
| 判别式 b² - 4ac | 根的情况 | 图像特征 |
| Δ > 0 | Two distinct real roots | Crosses x-axis twice |
| Δ = 0 | One repeated real root | Touches x-axis once |
| Δ < 0 | No real roots | Does not touch x-axis |
real roots; Δ = 0 means equal real roots; Δ < 0 means the equation has no real solutions.
牢记：Δ > 0 表示有两个不相等的实数根；Δ = 0 表示有两个相等的实数根；Δ < 0 表示方程无实数解。
7. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful for finding the vertex of a parabola and solving equations.
配方法可以将二次函数改写为 a(x + p)² + q 的形式。这对于求抛物线的顶点和求解方程特别有用。
x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4
Check: expand (x + 3)² - 4 = x² + 6x + 9 - 4 = x² + 6x + 5. ✓
验证：展开 (x + 3)² - 4 = x² + 6x + 9 - 4 = x² + 6x + 5，结果正确。
8. The Vertex and Line of Symmetry | 顶点与对称轴
ry is the vertical line x = h. The vertex is the maximum or minimum point of the parabola.
对于形式为 a(x - h)² + k 的二次函数，顶点坐标为 (h, k)，对称轴是竖直线 x = h。顶点就是抛物线的最高点或最低点。
For f(x) = ax² + bx + c: h = -b / 2a
Once you find h, substitute it back into the function to find k, the minimum or maximum value of f(x).
求出 h 后，将其代回原函数即可求出 k，也就是 f(x) 的最小值或最大值。
9. Transformations of Quadratic Graphs | 二次函数图像的变换
Quadratic graphs can be transformed by translations and reflections. Understanding these transformations helps you sketch graphs and interpret functions.
二次函数图像可以通过平移和翻折进行变换。理解这些变换有助于你画图和解读函数。
- y = f(x) + k: vertical translation upward (k > 0) or downward (k < 0).
- y = f(x) + k：图像沿 y 轴方向平移，k > 0 向上平移，k < 0 向下平移。
- y = f(x + h): horizontal translation left (h > 0) or right (h < 0).
- y = f(x + h)：图像沿 x 轴方向平移，h > 0 向左平移，h < 0 向右平移。
- y = -f(x): reflection across the x-axis.
- y = -f(x)：图像关于 x 轴翻折。
10. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities are solved by drawing the parabola and observing where the graph lies above or below the x-axis. Always sketch the graph first.
解二次不等式需要画出抛物线，并观察图像在 x 轴上方或下方的区间。解题时务必先画图。
x² - x - 6 < 0 → (x - 3)(x + 2) < 0
The roots are x = -2 and x = 3. The parabola opens upward, so the solution is -2 < x < 3.
方程的根为 x = -2 和 x = 3。抛物线开口向上，因此不等式的解集为 -2 < x < 3。
11. Applications in Problem Solving | 二次函数在应用题中的运用
Quadratic functions are widely used to model real-life situations such as projectile motion, area maximisation, and profit optimisation. IGCSE exam questions often connect quadratics with geometry or practical contexts.
二次函数常被用来模拟现实问题，如抛体运动、面积最大化和利润优化等。IGCSE 考试题经常将二次函数与几何或实际问题结合考查。
m, the area A = x(10 - x) = -x² + 10x. The maximum area occurs at x = 5, giving A = 25 m².
例如：一个矩形周长为 20 m。设一边长为 x m，则面积 A = x(10 - x) = -x² + 10x。面积在 x = 5 时达到最大值，即 A = 25 m²。
12. Common Exam Mistakes | 常见考试错误
Many students lose marks on quadratic questions due to avoidable errors. Here are the most frequent pitfalls to watch out for.
许多学生在二次函数题上因可以避免的错误而失分。以下是最常见的几类陷阱，务必警惕。
- Forgetting that a quadratic equation can have zero, one or two solutions.
- 忘记二次方程可能有零个、一个或两个解。
- Making sign errors when using the quadratic formula, especially with negative b values.
- 使用求根公式时符号出错，尤其是当 b 为负数时。
- Neglecting to set the equation to zero before using factorisation.
- 在使用因式分解前忘记将方程化为等于零的形式。
- Misreading the vertex when the function is written in standard form.
- 当函数以标准形式给出时，错误地判断顶点坐标。
Always check your answers by substituting the roots back into the original equation.
解出根后，务必将根代入原方程进行检验，以确保答案正确。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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