Quadratic Functions and Their Graphs | 二次函数及其图像

📚 Quadratic Functions and Their Graphs | 二次函数及其图像

A quadratic function is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, problem solving, and even in coordinate geometry. Understanding its forms, graph shape, and solving methods will help you tackle many exam questions confidently.

二次函数是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、应用题甚至坐标几何中。理解它的形式、图像形状和求解方法，将帮助你在考试中自信地处理许多问题。

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

A quadratic function in one variable is any expression that can be written in the form:

f(x) = ax² + bx + c, where a ≠ 0

Here, \(a\) is the coefficient of \(x²\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. The condition \(a ≠ 0\) is essential; if \(a = 0\), the function becomes linear, not quadratic.

这里，\(a\) 是 \(x²\) 的系数，\(b\) 是 \(x\) 的系数，\(c\) 是常数项。条件 \(a ≠ 0\) 至关重要；如果 \(a = 0\)，函数就变成一次函数，而不是二次函数。

The graph of a quadratic function is called a parabola. The sign of \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

二次函数的图像称为抛物线。\(a\) 的正负决定抛物线开口向上（\(a > 0\)）还是开口向下（\(a < 0\)）。

2. The Graph of y = ax² + bx + c | y = ax² + bx + c 的图像

When you plot a quadratic function, the shape is always a parabola. For example, \(y = x²\) is the simplest upward-opening parabola with its vertex at the origin (0, 0).

当你绘制二次函数图像时，形状总是一条抛物线。例如，\(y = x²\) 是最简单的开口向上的抛物线，顶点在原点 (0, 0)。

Changing \(a\) affects the width and direction of the parabola. A larger \(|a|\) makes the graph narrower; a smaller \(|a|\) makes it wider.

改变 \(a\) 会影响抛物线的宽度和方向。\(|a|\) 越大，图像越窄；\(|a|\) 越小，图像越宽。

The constant term \(c\) represents the y-intercept, that is, the point where the graph crosses the y-axis. At this point \(x = 0\), so \(y = c\).

常数项 \(c\) 表示 y 轴截距，即图像与 y 轴相交的点。在该点上 \(x = 0\)，所以 \(y = c\)。

3. Vertex and Axis of Symmetry | 顶点和对称轴

Every parabola has a vertical line of symmetry. This line passes through the vertex, the highest or lowest point of the graph.

每条抛物线都有一条竖直对称轴。这条线经过顶点，也就是图像的最高点或最低点。

For a quadratic function written as \(y = ax² + bx + c\), the axis of symmetry is given by the equation:

x = −b / (2a)

The x-coordinate of the vertex is also \(−b/(2a)\). To find the y-coordinate, substitute this value back into the original equation.

顶点的 x 坐标也是 \(−b/(2a)\)。要找到 y 坐标，将这个值代回原方程即可。

If \(a > 0\), the vertex is a minimum point; if \(a < 0\), the vertex is a maximum point.

如果 \(a > 0\)，顶点是最低点；如果 \(a < 0\)，顶点是最高点。

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

A quadratic equation has the form \(ax² + bx + c = 0\). One common solving method is factorisation, which works when the expression can be written as a product of two linear factors.

二次方程的形式为 \(ax² + bx + c = 0\)。一种常见的解法是因式分解，当表达式可以写成两个一次因式的乘积时适用。

For example, solve \(x² − 5x + 6 = 0\). We look for two numbers that multiply to 6 and add to −5. These are −2 and −3, so:

例如，解 \(x² − 5x + 6 = 0\)。我们寻找两个数，它们相乘等于 6，相加等于 −5。这两个数是 −2 和 −3，所以：

(x − 2)(x − 3) = 0

Setting each factor to zero gives \(x = 2\) or \(x = 3\). Always check your solutions by substituting them back into the original equation.

令每个因式等于零，得到 \(x = 2\) 或 \(x = 3\)。务必通过代回原方程来检查你的解。

Some quadratics are not easily factorised, especially when the coefficient of \(x²\) is not 1. In those cases, other methods are needed.

有些二次式不易因式分解，尤其是当 \(x²\) 的系数不为 1 时。在这种情况下，需要其他方法。

5. The Quadratic Formula | 二次公式

The quadratic formula is a universal method for solving any quadratic equation. For \(ax² + bx + c = 0\):

二次公式是解任何二次方程的通用方法。对于 \(ax² + bx + c = 0\)：

x = (−b ± √(b² − 4ac)) / (2a)

The symbol ± means that there are usually two solutions: one with a plus sign and one with a minus sign. This formula works for all cases, including when the roots are rational, irrational, or even non-real.

符号 ± 表示通常有两个解：一个用加号，一个用减号。这个公式适用于所有情况，包括根为有理数、无理数、甚至不是实数的情况。

You must memorise this formula for the IGCSE examination. Practice using it with different values of \(a\), \(b\), and \(c\) to gain fluency.

你必须为 IGCSE 考试记住这个公式。练习使用不同的 \(a\)、\(b\) 和 \(c\) 值，以熟练掌握。

Example: Solve 2x² − 4x − 3 = 0 using the formula.

Here \(a = 2\), \(b = −4\), \(c = −3\). Substituting into the formula gives:

这里 \(a = 2\)，\(b = −4\)，\(c = −3\)。代入公式得到：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = 1 ± ½√10

So the solutions are \(x ≈ 2.581\) and \(x ≈ −0.581\).

所以解为 \(x ≈ 2.581\) 和 \(x ≈ −0.581\)。

6. Completing the Square | 配方法

Completing the square is another important technique. It rewrites a quadratic expression in the form \(a(x + p)² + q\), which directly reveals the vertex.

配方法是另一项重要技巧。它将二次表达式改写为 \(a(x + p)² + q\) 的形式，直接显示出顶点。

To complete the square for \(x² + bx + c\), add and subtract \((b/2)²\) inside the expression. For example, for \(x² + 6x + 5\):

要对 \(x² + bx + c\) 配方，在表达式中加上并减去 \((b/2)²\)。例如，对于 \(x² + 6x + 5\)：

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

From this form, we see that the vertex is at \((−3, −4)\) and the axis of symmetry is \(x = −3\).

从这种形式，我们看到顶点在 \((−3, −4)\)，对称轴是 \(x = −3\)。

When the coefficient \(a ≠ 1\), first factor out \(a\) from the \(x²\) and \(x\) terms, then complete the square inside the brackets.

当系数 \(a ≠ 1\) 时，先从 \(x²\) 和 \(x\) 项中提出 \(a\)，再在括号内配方。

7. The Discriminant | 判别式

The discriminant is the expression inside the square root in the quadratic formula: \(b² − 4ac\). It tells us how many real roots a quadratic equation has without solving it.

判别式是二次公式根号内的表达式：\(b² − 4ac\)。它告诉我们二次方程有多少个实数根，而无需解方程。

If \(b² − 4ac > 0\), there are two distinct real roots.

如果 \(b² − 4ac > 0\)，有两个不同的实数根。
If \(b² − 4ac = 0\), there is exactly one real root (a repeated root).

如果 \(b² − 4ac = 0\)，恰好有一个实数根（重根）。
If \(b² − 4ac < 0\), there are no real roots; the roots are complex.

如果 \(b² − 4ac < 0\)，没有实数根；根为复数。

Graphically, this corresponds to the parabola crossing the x-axis twice, touching it once, or not meeting it at all.

在图像上，这分别对应抛物线与 x 轴相交两次、相切一次，或者完全不相交。

8. Sketching Quadratic Graphs | 绘制二次函数图像草图

To sketch a quadratic graph, you need to identify four key features: the y-intercept, the x-intercepts (if any), the vertex, and the direction of opening.

要绘制二次函数图像的草图，你需要确定四个关键特征：y 截距、x 截距（如果有）、顶点以及开口方向。

The y-intercept is simply the point \((0, c)\). The x-intercepts are the roots of the equation \(ax² + bx + c = 0\). The vertex can be found by completing the square or using the formula \(x = −b/(2a)\).

y 截距就是点 \((0, c)\)。x 截距是方程 \(ax² + bx + c = 0\) 的根。顶点可以通过配方或使用公式 \(x = −b/(2a)\) 求得。

Once you have these points, plot them and draw a smooth curve through them. Remember that the parabola is symmetrical about its axis of symmetry.

一旦有了这些点，就可以画出它们并连成平滑曲线。记住抛物线关于其对称轴对称。

Example: Sketch y = x² − 2x − 3.

Here \(a = 1 > 0\), so it opens upward. The y-intercept is \((0, −3)\). Solving \(x² − 2x − 3 = 0\) gives \(x = 3\) or \(x = −1\). The vertex x-coordinate is \(−(−2)/(2) = 1\), and \(y = 1 − 2 − 3 = −4\), so the vertex is \((1, −4)\).

这里 \(a = 1 > 0\)，所以开口向上。y 截距是 \((0, −3)\)。解 \(x² − 2x − 3 = 0\) 得到 \(x = 3\) 或 \(x = −1\)。顶点 x 坐标是 \(−(−2)/(2) = 1\)，且 \(y = 1 − 2 − 3 = −4\)，所以顶点是 \((1, −4)\)。

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

Quadratic functions appear in many real-world situations, such as projectile motion, area optimisation, and profit maximisation.

二次函数出现在许多现实情境中，例如抛体运动、面积优化和利润最大化。

For example, the height of a ball thrown upward can be modelled as \(h(t) = −5t² + 20t + 2\), where \(t\) is time in seconds. To find the maximum height, locate the vertex.

例如，向上抛出的球的高度可以建模为 \(h(t) = −5t² + 20t + 2\)，其中 \(t\) 是以秒为单位的时间。要找到最大高度，需要找到顶点。

Using \(t = −b/(2a) = −20/(2 × −5) = 2\) seconds, then \(h(2) = −20 + 40 + 2 = 22\) metres. So the ball reaches a maximum height of 22 metres after 2 seconds.

使用 \(t = −b/(2a) = −20/(2 × −5) = 2\) 秒，然后 \(h(2) = −20 + 40 + 2 = 22\) 米。因此球在 2 秒后达到最大高度 22 米。

When solving word problems, define the variable clearly, form a quadratic equation, solve it using a suitable method, and check whether the solution makes sense in the context.

解决文字题时，要明确定义变量，建立二次方程，用合适的方法求解，并检查解在上下文中是否合理。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

One common mistake is forgetting that \(a\) must not be zero. Another is misidentifying the sign of \(b\) in the quadratic formula when \(b\) itself is negative.

一个常见错误是忘记 \(a\) 不能为零。另一个是在二次公式中，当 \(b\) 本身为负数时，弄错 \(b\) 的符号。

Always simplify roots fully. For example, \(\sqrt{40}\) should be written as \(2\sqrt{10}\). In the quadratic formula, remember to divide both the numerator terms by \(2a\).

始终要彻底化简根号。例如，\(\sqrt{40}\) 应写成 \(2\sqrt{10}\)。在二次公式中，记得分子两项都要除以 \(2a\)。

When sketching a graph, always show key points clearly: the y-intercept, x-intercepts, and vertex. Label the axis of symmetry if possible.

画图时，要清晰地标出关键点：y 截距、x 截距和顶点。如果可能，标出对称轴。

Use the discriminant quickly to determine the number of roots before attempting to solve. This saves time and prevents unnecessary calculation errors.

在尝试求解之前，先用判别式快速确定根的个数。这可以节省时间并避免不必要的计算错误。

Finally, practise past paper questions. Quadratics are a favourite topic in IGCSE exams, and regular practice will make you familiar with all the common variations.

最后，多练历年真题。二次函数是 IGCSE 考试中的热门主题，定期练习会让你熟悉所有常见的变化题型。

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Quadratic Functions and Their Graphs | 二次函数及其图像