📚 158. Quadratic Graphs | 二次函数图像
x + c (where a ≠ 0), and its graph is a smooth, symmetric curve called a parabola. Being able to plot, interpret and transform these graphs is essential for success on the Edexcel IGCSE exam, as questions often involve vertex coordinates, intercepts, solving equations graphically and sketching. This revision guide covers every key concept, step by step, to help you master quadratic graphs.
在IGCSE数学中，二次函数图像是理解诸多代数与图形问题的基础。二次函数的一般形式为 y = ax² + bx + c（其中 a ≠ 0），其图像是一条光滑、对称的曲线，称为抛物线。能够正确绘制、解读并变换这些图像对于在Edexcel IGCSE考试中取得成功至关重要，因为相关题目常常涉及顶点坐标、截距、利用图像解方程以及草图绘制。本复习指南将逐步覆盖每一个关键概念，帮助你彻底掌握二次函数图像。
1. Understanding Quadratic Functions | 理解二次函数
stants and a is not zero. The highest power of x is 2, which is why it is called a quadratic. On a graph, the term ax² dominates the shape, giving the characteristic U-shaped or ∩ coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
二次函数是指可以写成标准形式 y = ax² + bx + c 的任何函数，其中 a、b、c 为常数且 a 不为零。因为 x 的最高次幂是 2，所以称为二次函数。在图像上，ax² 项主导了形状，形成特有的 U 形或 ∩ 形曲线，即抛物线。系数 a 决定了抛物线开口向上（a > 0）还是向下（a < 0）。
and has its vertex at (0,0). From this parent function, all other quadratic graphs can be considered as transformations involving stretches, reflections and translations. Understanding the basic curve is the first step towards mastering more complex examples.
最简单的二次函数是 y = x²，它经过原点且顶点在 (0,0)。从这个母函数出发，所有其他二次函数图像都可以视为通过伸缩、反射和平移等变换得到。理解这条基本曲线是掌握更复杂例子的第一步。
2. The Shape of a Quadratic Graph | 二次函数图像的形状
Every quadratic graph is a parabola, and its shape depends on the coefficient a. If a is positive, the parabola has a minimum point and looks like a smiling face (∪). If a is negative, the parabola has a maximum point and looks like a frowning face (∩). The magnitude of a also controls how wide or narrow the parabola appears; the larger |a| is, the narrower the curve, and the smaller |a| is, the wider the curve.
每一个二次函数图像都是一条抛物线，其形状取决于系数 a。若 a 为正，抛物线有一个最小值点，形状像笑脸（∪）；若 a 为负，抛物线有一个最大值点，形状像一个皱眉的脸（∩）。a 的绝对值大小还控制着抛物线的宽窄：|a| 越大，曲线越窄；|a| 越小，曲线越宽。
ram with the correct orientation and to check a few plotted points to ensure the curve is smooth, not made up of straight lines.
例如，y = 2x² 比 y = x² 更窄，而 y = ½x² 更宽。这种伸缩效应对于所有形如 y = ax² 的抛物线都是关于 y 轴对称的。在图上标出正确的开口方向并要求画出的曲线是平滑的，而非由直线段拼接而成，这一点很重要。
3. Plotting a Quadratic Graph | 绘制二次函数图像
o calculate the corresponding y-values. Plot the points on a set of axes, and then draw a smooth curve passing through all points. Do not join the dots with straight lines — a parabola must be a smooth, continuous curve.
要绘制二次函数图像，先从填制一个数值表开始。选择能够覆盖顶点和两个根的 x 值范围。将每个 x 值代入方程 y = ax² + bx + c，计算出对应的 y 值。将点描在坐标轴上，然后画一条光滑的曲线穿过所有点。不要用直线连接各点——抛物线必须是一条光滑、连续的曲线。
When completing a table of values, pay attention to symmetry. For a quadratic function, the y-values on either side of the vertex are mirror images. This can be used as a check: if you find the values are not symmetric, you may have made a calculation error. A well-drawn curve will also allow you to read off approximate solutions to related equations.
填制数值表时要注意对称性。对于二次函数，顶点两侧的 y 值互为镜像。可以用这一特性来检查：如果发现数值不对称，则可能是计算出了错。画得精准的曲线还能让你读出相关方程的近似解。
4. The Axis of Symmetry | 对称轴
line passes through the vertex, meaning the x-coordinate of the vertex is exactly -b/(2a).
对称轴是一条将抛物线分成两个镜像半边的垂直线。对于标准形式 y = ax² + bx + c 的二次函数，对称轴总是由公式 x = -b / (2a) 给出。这条线经过顶点，也就是说顶点的 x 坐标正是 -b/(2a)。
x = -b / (2a)
Knowing the axis of symmetry is extremely useful when sketching. You can quickly locate the vertex by plugging this x-value back into the original equation to find the maximum or minimum y-value. The axis of symmetry also helps to predict the shape of the graph without plotting dozens of points.
了解对称轴在画草图时极其有用。将这个 x 值回代入原方程，你就可以快速求得最大或最小的 y 值，从而定位顶点。对称轴还能帮助你在不逐点描画的情况下预测图像形状。
5. Finding the Vertex | 求顶点
-b / (2a), then substitute this x-value into y = ax² + bx + c to obtain the y-coordinate.
顶点是抛物线的转折点。对于正二次函数（a > 0），顶点是最低点（最小值）；对于负二次函数（a < 0），它是最高点（最大值）。顶点坐标可以利用对称轴求得：先计算 x = -b / (2a)，然后将这个 x 值代入 y = ax² + bx + c 得到 y 坐标。
s the vertex. Both methods are acceptable, but using -b/(2a) is often quicker when the function is given in standard form. The vertex is crucial for identifying the range of the function and solving optimisation problems.
另一种方法是通过完成平方将二次函数重新写成 y = a(x - h)² + k 的形式，此时 (h, k) 就是顶点。两种方法皆可，但当函数以标准形式给出时，使用 -b/(2a) 通常更快。顶点对于确定函数的取值范围以及解决优化问题至关重要。
6. Intercepts with the Axes | 与坐标轴的截距
which can be done by factorising, using the quadratic formula, or reading from the graph.
二次函数图像的 y 轴截距可通过设 x = 0 求得。这总是得到点 (0, c)，因为 y = a(0)² + b(0) + c = c。这是最容易描画的点，也是整条曲线的参照点。x 轴截距（也称根或零点）通过解方程 ax² + bx + c = 0 求得，可以用因式分解、二次公式或从图像上直接读取。
If the discriminant b² - 4ac is positive, there are two distinct x-intercepts. If it is zero, the graph touches the x-axis at exactly one point (the vertex). If it is negative, the graph does not cross the x-axis at all. These cases are vital for understanding the number of real solutions to quadratic equations.
如果判别式 b² - 4ac 为正，则有两个不同的 x 轴截距；若为零，则图像恰好与 x 轴相切于一点（即顶点）；若为负，则图像完全不与 x 轴相交。这些情况对于理解二次方程的实数解个数十分关键。
7. Solving Quadratic Equations Graphically | 利用图像解二次方程
. This visual method is particularly helpful when the quadratic does not factorise easily.
可以通过画出抛物线 y = ax² + bx + c 并找出曲线与 x 轴交点的 x 坐标来图解二次方程。这些 x 值就是方程 ax² + bx + c = 0 的解。这种直观的方法在二次式不易因式分解时尤其有用。
, and look for intersections. This technique is often tested in IGCSE, so it is essential to understand how to rearrange the equation appropriately.
你还可以在同一坐标系中画一条合适的直线来图解更复杂的方程。例如，要解 x² - 3x - 1 = 0，可以画出 y = x² - 3x，再画出 y = 1，然后寻找交点。这一技巧在IGCSE考试中经常出现，因此理解如何恰当地对方程进行变形至关重要。
8. Sketching Quadratics from Factorised Form | 由因式分解形式画草图
t x = (p + q)/2. Substituting this into the equation gives the y-coordinate of the vertex.
当二次函数以因式分解形式 y = (x - p)(x - q) 给出时，你可以立即确定 x 轴截距：它们位于 x = p 和 x = q 处。若 x² 前的系数为正（此处为 1），则图像开口向上。对称轴恰好位于两个根的中间，即 x = (p + q)/2。将此值代入方程即可得到顶点的 y 坐标。
e parabola upside down. Factorised form is one of the most efficient ways to handle exam questions involving graphs of quadratic functions.
这种方法可以实现快速画草图：描出两个根，确定顶点，然后通过这些三点画一条对称的曲线。如果二次函数前面有一个负号，比如 y = -(x - p)(x - q)，只需将抛物线上下翻转。因式分解形式是应对涉及二次函数图像考题的最有效方式之一。
9. Effect of the Coefficient a | 系数 a 的影响
rols both the direction of opening and the steepness of the parabola. When a > 0, the parabola opens upwards and has a minimum point. When a < 0, the parabola opens downwards and has a maximum point. As |a| increases, the graph becomes narrower (more steep). As |a| decreases below 1, the graph becomes wider (flatter).
y = ax² + bx + c 中的系数 a 既控制抛物线的开口方向，又控制其陡峭程度。当 a > 0 时，抛物线开口向上，有一个最小值点；当 a < 0 时，抛物线开口向下，有一个最大值点。随着 |a| 增大，图像变得越来越窄（更陡）；当 |a| 减小到 1 以下时，图像变宽（更平）。
as a vertical stretch factor from the x-axis. Recognizing these changes without a full table of values is a key skill for speed and accuracy.
此外，a 会显著影响给定 x 值对应的 y 值。比较 y = x²、y = 2x² 和 y = ½x² 的数值表可以清晰地看出这一点。如果二次函数写成 y = a(x - h)² + k 的形式，那么 a 还起到从 x 轴方向进行竖向伸缩的作用。能在不依赖完整数据表的情况下识别这些变化，是兼顾速度与准确性的关键技能。
10. Transformations of y = x² | y = x² 的变换
ifts the graph 3 units to the right. A translation up or down is achieved by adding a constant k: y = x² + 2 shifts the graph 2 units up.
所有二次函数图像都是基本抛物线 y = x² 的变换。主要有三种变换类型：平移（移动）、反射（翻转）和伸缩（纵向缩放）。左右平移可通过将 x 替换为 (x - h) 来实现：例如，y = (x - 3)² 将图像向右平移 3 个单位。上下平移可通过加上常数 k 来实现：y = x² + 2 将图像向上平移 2 个单位。
, allows you to sketch any quadratic quickly by starting from y = x² and applying the transformations step by step.
关于 x 轴的反射可将整个表达式变负来实现：y = -x² 将抛物线上下翻转。纵向伸缩可在前面加上系数：y = 3x² 将图像纵向拉伸为原来的 3 倍。将这些变换组合起来，如 y = -2(x + 1)² - 4，你就可以从 y = x² 出发，逐步应用变换，从而快速画出任意二次函数的图像。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图