📚 Quadratic Graphs Made Simple | 轻松掌握二次函数图像
Quadratic functions appear throughout the IGCSE Mathematics syllabus. Understanding their graphs is essential for solving equations, interpreting real-world situations, and scoring well in exams. This article breaks down every key skill you need, with clear explanations and step-by-step reasoning.
二次函数在 IGCSE 数学考纲中反复出现。掌握它们的图像，是解方程、理解实际场景以及在考试中取得高分的关键。本文将逐一拆解所有必备技能，提供清晰讲解与逐步推理。
1. Standard Form of a Quadratic | 二次函数的标准式
The general form of a quadratic function is y = ax2+bx+c ² + bx + c, where a, b and c are constants and a ≠ 0. The graph of this function is a curve called a parabola.
二次函数的一般式为 y = ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。它的图像是一条抛物线。
The value of a determines whether the parabola opens upward or downward. If a > 0, it opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.
a 的正负决定抛物线的开口方向。若 a > 0，开口向上，存在最低点；若 a < 0，开口向下，存在最高点。
- If a is large, the parabola is narrow; if a is small, it is wide. | a 的绝对值越大，抛物线越窄；绝对值越小，抛物线越宽。
- The constant c is the y-intercept, where the graph crosses the y-axis. | 常数 c 是 y 轴截距，即图像与 y 轴的交点。
2. The Axis of Symmetry | 对称轴
Every parabola has a vertical line of symmetry. This line passes through the vertex (the maximum or minimum point) and divides the graph into two mirror images. The equation of the axis of symmetry is found using the coefficients a and b.
每条抛物线都有一条竖直对称轴。该直线经过顶点（最高点或最低点），将图像分成左右镜像的两部分。对称轴的方程由系数 a 和 b 确定。
x = -b / (2a)
-4. So the axis is x = -(-4) / (2×2) = 4 / 4 = 1. The line x = 1 is the axis of symmetry.
例如，对于 y = 2x² - 4x + 1，a = 2，b = -4。因此对称轴为 x = -(-4) / (2×2) = 4 / 4 = 1，即直线 x = 1。
3. The Vertex | 顶点
/ (2a). To find the y-coordinate, substitute this x-value back into the original equation.
顶点是抛物线的转折点。其 x 坐标与对称轴相同：x = -b / (2a)。要求 y 坐标，只需将 x 值代回原方程。
1 = 2 - 4 + 1 = -1. So the vertex is at (1, -1). Since a = 2 > 0, this is a minimum point.
仍用上例，y = 2(1)² - 4(1) + 1 = 2 - 4 + 1 = -1。因此顶点为 (1, -1)。因为 a = 2 > 0，所以这是最低点。
- Complete the square to write y = a(x - h)² + k; the vertex is (h, k). | 配方法可将 y = a(x - h)² + k，顶点即为 (h, k)。
- When a > 0, the vertex is the minimum; when a < 0, it is the maximum. | 当 a > 0 时顶点为最小值；当 a < 0 时顶点为最大值。
4. Roots of a Quadratic Equation | 二次方程的根
ues where the graph crosses the x-axis. Graphically, these are the x-intercepts. You can find them by factorising, using the quadratic formula, or completing the square.
二次方程 ax² + bx + c = 0 的根（或解）是图像与 x 轴交点的 x 坐标，即 x 轴截距。求根的方法包括因式分解、使用求根公式或配方法。
x = (-b ± √b² - 4ac) / (2a)
The quadratic formula works for all quadratics. The expression inside the square root, b² - 4ac, is called the discriminant. It tells us how many real roots there are.
求根公式适用于所有二次方程。根号内的表达式 b² - 4ac 称为判别式，它决定了实根的个数。
- If b² - 4ac > 0, there are two distinct real roots. | 若 b² - 4ac > 0，有两个不同的实根。
- If b² - 4ac = 0, there is exactly one real root (a repeated root). | 若 b² - 4ac = 0，有一个实根（重根）。
- If b² - 4ac < 0, there are no real roots. | 若 b² - 4ac < 0，没有实根。
5. The y-intercept | y 轴截距
Substituting x = 0 into y = ax² + bx + c gives y = c. So the y-intercept is simply (0, c).
y 轴截距是图像与 y 轴的交点。该点总是出现在 x = 0 处。将 x = 0 代入 y = ax² + bx + c，得到 y = c。因此 y 轴截距就是 (0, c)。
When sketching a quadratic graph, you should always mark the y-intercept clearly. It also helps to check whether a sketch is reasonable if one root is known.
画二次函数草图时，务必标出 y 轴截距。当已知一个根时，它也能帮助检查草图是否合理。
6. Sketching a Quadratic Graph | 画二次函数草图
To sketch a quadratic graph, follow these steps. First, identify whether it opens upward or downward using the sign of a. Then find the y-intercept, the axis of symmetry, the vertex, and the roots if they exist.
画二次函数草图需按以下步骤：首先，根据 a 的符号判断开口方向；然后求 y 轴截距、对称轴、顶点，以及根（若存在）。
- Mark the y-intercept at (0, c). | 标记 y 轴截距 (0, c)。
- Draw the axis of symmetry as a dashed line. | 用虚线画出对称轴。
- Plot the vertex. | 标出顶点。
- If there are real roots, plot them on the x-axis. | 若有实根，在 x 轴上标出。
- Join the points with a smooth curve, symmetric about the axis. | 用平滑曲线连接各点，并保持关于对称轴对称。
If there are no real roots, the graph does not touch the x-axis. It may lie entirely above or below the x-axis, depending on the sign of a and the y-intercept.
若没有实根，图像与 x 轴不相交。它可能完全位于 x 轴上方或下方，具体取决于 a 与 y 轴截距的符号。
7. Transformations of Quadratic Graphs | 二次函数图像的变换
x - h)² + k, changing h moves the graph left or right, and changing k moves it up or down.
二次函数图像可以平移或翻转。在顶点式 y = a(x - h)² + k 中，改变 h 会使图像左右平移，改变 k 会使图像上下平移。
- Replacing x by (x - h) shifts the graph h units to the right if h > 0, and |h| units to the left if h < 0. | 将 x 替换为 (x - h)：当 h > 0 时图像右移 h 个单位；当 h < 0 时图像左移 |h| 个单位。
- Adding k outside the square shifts the graph upward by k units if k > 0, and downward by |k| if k < 0. | 在平方项外加 k：当 k > 0 时图像上移 k 个单位；当 k < 0 时图像下移 |k| 个单位。
- Replacing a by -a reflects the graph in the x-axis. | 将 a 变为 -a，图像关于 x 轴翻折。
These transformations are useful in exam questions that ask you to describe or sketch a graph from its equation.
这些变换在做“根据方程描述或画出图像”的考试题时非常有用。
8. Using the Discriminant | 判别式的应用
t a number; it links directly to the graph. For a quadratic graph, it tells you whether the parabola crosses the x-axis, touches it once, or never reaches it.
判别式 Δ = b² - 4ac 不仅仅是一个数，它与图像直接相关。对于二次函数图像，它告诉你抛物线是与 x 轴相交于两点、相切于一点，还是完全不相交。
| Δ value | Graph interpretation | 中文解释 |
| Δ > 0 | Two x-intercepts | 两个 x 轴交点 |
| Δ = 0 | One x-intercept (vertex on axis) | 一个 x 轴交点（顶点在轴上） |
| Δ < 0 | No x-intercept | 没有 x 轴交点 |
atic equation has real roots. Set the discriminant ≥ 0 and solve the resulting inequality.
考试中可能要求你求 k 的范围，使得某个二次方程有实根。此时令判别式 ≥ 0，然后解所得不等式。
9. Quadratic Inequalities on a Graph | 用图像解二次不等式
solve x² - 3x + 2 > 0. First factorise: (x - 1)(x - 2) > 0. The roots are x = 1 and x = 2.
当你画出或想象出二次函数图像后，就可以轻松解二次不等式。例如，解 x² - 3x + 2 > 0。先因式分解：(x - 1)(x - 2) > 0。根为 x = 1 和 x = 2。
ens upward. It is above the x-axis when x < 1 or x > 2. So the solution is x < 1 or x > 2.
因为 a = 1 > 0，抛物线开口向上。当 x < 1 或 x > 2 时，图像在 x 轴上方。因此解为 x < 1 或 x > 2。
- For ax² + bx + c > 0 with a > 0, the solution is outside the interval between the roots. | 当 a > 0 且 ax² + bx + c > 0 时，解在两根之外。
- For ax² + bx + c < 0 with a > 0, the solution is between the roots. | 当 a > 0 且 ax² + bx + c < 0 时，解在两根之间。
10. Solving Word Problems with Quadratics | 二次函数应用题
t) = -5t² + 20t + 2. The graph opens downward, so the maximum height occurs at the vertex.
许多实际问题涉及二次函数图像。例如，向上抛出的球的高度 h(t) 可表示为 h(t) = -5t² + 20t + 2。图像开口向下，因此最大高度出现在顶点处。
a) = -20 / (2×(-5)) = 2 seconds. Then h(2) = -5(4) + 20(2) + 2 = -20 + 40 + 2 = 22 metres.
求最大高度：t = -b / (2a) = -20 / (2×(-5)) = 2 秒。然后 h(2) = -5(4) + 20(2) + 2 = -20 + 40 + 2 = 22 米。
t) = 0 and solve the quadratic equation. Only a positive root makes sense in this context.
应用题常问球何时落地。令 h(t) = 0 并解二次方程。在此情境下只有正根有意义。
11. Common Mistakes to Avoid | 常见易错点
ber the formula x = -b / (2a). If b is negative, the negative sign changes it to positive.
学生常在使用对称轴公式时弄错 b 的符号。记住公式 x = -b / (2a)。若 b 为负数，负负得正。
- Do not forget to include the minus sign in the quadratic formula. | 使用求根公式时不要遗漏负号。
- Do not confuse the meaning of c with the roots; c is the y-intercept, not an x-intercept. | 不要把 c 与根混淆；c 是 y 轴截距，不是 x 轴截距。
- When sketching, always check that the curve is symmetric. | 画草图时，务必检查曲线是否对称。
- When solving inequalities, note whether the endpoints are included: use ≤, ≥ vs <, >. | 解不等式时注意端点是否取等：≤、≥ 与 <、> 不同。
12. Summary of Key Formulas | 关键公式总结
Here are the essential formulas for quadratic graphs. Commit them to memory, but also understand how they are derived. This understanding will help you in unfamiliar exam questions.
以下是二次函数图像的关键公式。请熟记它们，但也要理解它们的推导过程。这将帮助你在陌生题型中灵活运用。
| Item | Formula | 中文表述 |
| Axis of symmetry | x = -b / (2a) | 对称轴 x = -b / (2a) |
| Vertex | ( -b/(2a), f(-b/(2a)) ) | 顶点坐标 |
| Quadratic formula | x = (-b ± √b² - 4ac) / (2a) | 求根公式 |
| Discriminant | Δ = b² - 4ac | 判别式 |
| Completed square | y = a(x - h)² + k | 配方法后顶点式 |
By mastering these skills, you can confidently handle any quadratic graph question on the IGCSE exam. Practice by drawing graphs, solving equations, and checking your sketches with a calculator or graphing software.
掌握了这些技能，你就能自信应对 IGCSE 考试中任何关于二次函数图像的题目。请通过画图、解方程，并用计算器或绘图软件检查草图来多加练习。
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