Mastering Quadratic Functions & Graphs | 掌握二次函数与图像

📚 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

Notice the symmetry: f(-1) = f(3) = 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

The roots (or solutions) of a quadratic equation are the x-values where the graph crosses the x-axis, i.e., where f(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

Remember: Δ > 0 means two different 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

For a quadratic in the form a(x – h)² + k, the vertex is at the point (h, k). The line of symmetry 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。

For example: a rectangle has a perimeter of 20 m. If one side is x 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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Mastering Quadratic Functions & Graphs | 掌握二次函数与图像