Graphs of Quadratic Functions | 二次函数图象

📚 Graphs of Quadratic Functions | 二次函数图象

A quadratic function is one of the most important topics in IGCSE Mathematics. It appears in almost every exam paper, either as a straightforward plotting question or as a problem-solving question involving roots, turning points, and symmetry. In this revision guide, we will go through everything you need to know about the graph of a quadratic function, step by step.

二次函数是 IGCSE 数学中最重要的知识点之一。几乎每一份考卷都会出现二次函数，有时是直接要求画图，有时是通过求根、顶点和对称轴来考查解题能力。在这篇复习指南中，我们将一步步梳理二次函数图象的所有核心内容。

1. The General Form and Shape | 一般形式与图象形状

A quadratic function can be written in the general form \( y = ax^2 + bx + c \), where \( a \), \( b \) and \( c \) are constants and \( a \neq 0 \). The graph of a quadratic function is always a curve called a parabola.

二次函数的一般形式为 \( y = ax^2 + bx + c \)，其中 \( a \)、\( b \)、\( c \) 为常数，且 \( a \neq 0 \)。二次函数的图象总是一条称为抛物线的曲线。

The sign of the coefficient \( a \) determines the direction in which the parabola opens. If \( a > 0 \), the parabola opens upwards and has a minimum point. If \( a < 0 \), the parabola opens downwards and has a maximum point.

二次项系数 \( a \) 的正负决定了抛物线的开口方向。当 \( a > 0 \) 时，抛物线开口向上，存在最小值点；当 \( a < 0 \) 时，抛物线开口向下，存在最大值点。

\( a > 0 \): U-shaped curve, minimum point
\( a < 0 \): N-shaped curve, maximum point

Remember this simple rule: a positive \( a \) gives a smile, a negative \( a \) gives a frown. This is the very first thing you should check when sketching a quadratic graph.

记住这个简单规律：\( a \) 为正时图象像微笑，\( a \) 为负时图象像皱眉。画二次函数草图时，首先要检查的就是这一点。

2. The Roots or x-intercepts | 根与 x 轴交点

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are the values of \( x \) where the graph crosses the x-axis. These are also called the x-intercepts of the parabola. A quadratic equation can have two distinct roots, one repeated root, or no real roots.

二次方程 \( ax^2 + bx + c = 0 \) 的根就是图象与 x 轴交点的横坐标，也称为抛物线的 x 轴截距。二次方程可以有两个不同实根、一个重根，或者没有实数根。

To find the roots, you can use factorisation, the quadratic formula, or completing the square. The quadratic formula is particularly useful when the equation cannot be easily factorised.

求根可以采用因式分解、公式法或配方法。当方程不容易因式分解时，公式法尤其有效。

\( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)

The expression \( b^2 – 4ac \) is called the discriminant. If \( b^2 – 4ac > 0 \), there are two distinct real roots. If \( b^2 – 4ac = 0 \), there is exactly one repeated root. If \( b^2 – 4ac < 0 \), there are no real roots and the graph does not intersect the x-axis.

表达式 \( b^2 – 4ac \) 称为判别式。当 \( b^2 – 4ac > 0 \) 时，方程有两个不同实根；当 \( b^2 – 4ac = 0 \) 时，有一个重根；当 \( b^2 – 4ac < 0 \) 时，没有实数根，图象与 x 轴没有交点。

Two roots: the parabola crosses the x-axis at two points.

两个根：抛物线与 x 轴相交于两点。
One root: the parabola touches the x-axis at the vertex.

一个根：抛物线在顶点处与 x 轴相切。
No roots: the parabola stays entirely above or below the x-axis.

没有根：抛物线完全位于 x 轴上方或下方。

3. The y-intercept | y 轴截距

The y-intercept of a quadratic graph is the point where the curve crosses the y-axis. This occurs when \( x = 0 \), so the y-intercept is simply the constant term \( c \) in the equation \( y = ax^2 + bx + c \).

二次函数图象与 y 轴的交点称为 y 轴截距。由于交点在 \( x = 0 \) 处，所以 y 轴截距就是方程 \( y = ax^2 + bx + c \) 中的常数项 \( c \)。

This is the easiest point to find on any quadratic graph. You simply read the value of \( c \) directly from the equation. For example, for \( y = 2x^2 – 3x + 5 \), the y-intercept is \( (0, 5) \).

这是二次函数图象上最容易找到的点，直接从方程中读出 \( c \) 的值即可。例如，对于 \( y = 2x^2 – 3x + 5 \)，y 轴截距为 \( (0, 5) \)。

When sketching a graph, always mark the y-intercept clearly. It serves as a reliable reference point and is often required in exam questions.

画草图时，一定要清楚地标出 y 轴截距。它是一个可靠的参考点，考试中也经常要求标出。

4. The Axis of Symmetry | 对称轴

Every parabola has a vertical line of symmetry that passes through its vertex. This line is called the axis of symmetry. For a quadratic function \( y = ax^2 + bx + c \), the equation of the axis of symmetry is

每条抛物线都有一条经过顶点的垂直对称线，称为对称轴。对于二次函数 \( y = ax^2 + bx + c \)，对称轴的方程为

\( x = -\frac{b}{2a} \)

The axis of symmetry is exactly halfway between the two roots, if they exist. This means that if you know the two roots \( x_1 \) and \( x_2 \), the axis of symmetry is simply the average: \( x = \frac{x_1 + x_2}{2} \).

如果两个根存在，对称轴正好位于两个根的正中间。也就是说，若已知两个根为 \( x_1 \) 和 \( x_2 \)，对称轴就是它们的平均值：\( x = \frac{x_1 + x_2}{2} \)。

For example, if the roots of a quadratic equation are \( x = 1 \) and \( x = 5 \), then the axis of symmetry is \( x = 3 \). This is a very useful shortcut in both sketching and problem-solving.

例如，若二次方程的两个根为 \( x = 1 \) 和 \( x = 5 \)，则对称轴为 \( x = 3 \)。这是画图和解题中非常有用的捷径。

5. The Vertex, Turning Point or Stationary Point | 顶点、极值点

The vertex of a parabola, also called the turning point, is the point where the graph changes direction. For an upward-opening parabola, the vertex is the minimum point. For a downward-opening parabola, the vertex is the maximum point.

抛物线的顶点，也称为极值点，是图象改变方向的点。对于开口向上的抛物线，顶点是最小值点；对于开口向下的抛物线，顶点是最大值点。

The x-coordinate of the vertex is the same as the axis of symmetry: \( x = -\frac{b}{2a} \). To find the y-coordinate, substitute this value of \( x \) back into the original equation.

顶点的 x 坐标与对称轴相同：\( x = -\frac{b}{2a} \)。要求 y 坐标，只需将该 \( x \) 值代回原方程即可。

Vertex: \( \left( -\frac{b}{2a}, \; f\left(-\frac{b}{2a}\right) \right) \)

Using the vertex form of a quadratic function makes this much easier. The vertex form is

使用二次函数的顶点式会让这个过程更加简单。顶点式为

\( y = a(x – h)^2 + k \)

In this form, the vertex is simply the point \( (h, k) \). The graph of \( y = a(x – h)^2 + k \) is the graph of \( y = ax^2 \) shifted horizontally by \( h \) units and vertically by \( k \) units.

在这种形式下，顶点就是点 \( (h, k) \)。函数 \( y = a(x – h)^2 + k \) 的图象是 \( y = ax^2 \) 的图象水平平移 \( h \) 个单位、垂直平移 \( k \) 个单位后得到的。

6. Completing the Square | 配方法

Completing the square is a powerful algebraic technique used to rewrite a quadratic function in vertex form. This allows you to read the vertex directly from the equation without substituting values.

配方法是一种强大的代数技巧，可以将二次函数改写为顶点式。这样可以直接从方程中读出顶点坐标，而不需要代值计算。

To complete the square for \( y = ax^2 + bx + c \), first factor out \( a \) from the first two terms, then add and subtract the square of half the coefficient of \( x \).

对 \( y = ax^2 + bx + c \) 配方时，先从第一项和第二项中提取 \( a \)，然后加上并减去 x 系数一半的平方。

\( x^2 + bx = \left(x + \frac{b}{2}\right)^2 – \left(\frac{b}{2}\right)^2 \)

For example, consider \( y = x^2 – 6x + 11 \). We can rewrite it as \( y = (x – 3)^2 + 2 \). Therefore, the vertex is \( (3, 2) \) and the axis of symmetry is \( x = 3 \).

例如，考虑 \( y = x^2 – 6x + 11 \)。我们可以将其改写为 \( y = (x – 3)^2 + 2 \)。因此，顶点为 \( (3, 2) \)，对称轴为 \( x = 3 \)。

Completing the square also helps in solving quadratic equations when factorisation is not possible, and it is essential for finding the minimum or maximum value of a quadratic expression.

当因式分解无法进行时，配方法也有助于求解二次方程，并且在求二次表达式的最小值或最大值时必不可少。

7. Sketching a Quadratic Graph | 画二次函数草图

Sketching a quadratic graph is a systematic process. Follow these steps to produce a clear and accurate sketch in any exam.

画二次函数草图是一个系统化的过程。按照以下步骤，你可以在考试中画出清晰准确的草图。

Step 1: Identify the sign of \( a \) to determine the shape.

第一步：判断 \( a \) 的正负，确定开口方向。
Step 2: Find the y-intercept \( (0, c) \).

第二步：找 y 轴截距 \( (0, c) \)。
Step 3: Find the roots by solving \( ax^2 + bx + c = 0 \), if they exist.

第三步：解方程 \( ax^2 + bx + c = 0 \) 求根（如果存在）。
Step 4: Find the axis of symmetry using \( x = -\frac{b}{2a} \).

第四步：用 \( x = -\frac{b}{2a} \) 求对称轴。
Step 5: Find the vertex by substituting the axis of symmetry into the equation.

第五步：将对称轴的 \( x \) 值代入方程求顶点。
Step 6: Plot all key points and draw a smooth curve through them.

第六步：标出所有关键点，并画一条平滑曲线穿过它们。

Always label the graph with its equation and mark all intercepts and the vertex clearly. A well-labelled sketch earns full marks even if the curve is not perfectly drawn.

始终在图象上标出方程，并清楚标记所有截距和顶点。即使曲线画得不够完美，只要标注齐全，也能拿到满分。

8. The Discriminant and the Nature of Roots | 判别式与根的性质

The discriminant \( \Delta = b^2 – 4ac \) tells us the number and nature of the roots without actually solving the equation. This is one of the most frequently tested ideas in IGCSE papers.

判别式 \( \Delta = b^2 – 4ac \) 可以在不解方程的情况下告诉我们根的个数和性质。这是 IGCSE 考试中最常考的内容之一。

Discriminant	Nature of roots	Graph intersection
\( \Delta > 0 \)	Two distinct real roots	Crosses x-axis at two points
\( \Delta = 0 \)	One repeated real root	Touches x-axis at one point
\( \Delta < 0 \)	No real roots	Does not touch x-axis

The discriminant is also used in problems that ask “find the range of values of \( k \) for which the equation has two real roots”. In such cases, you set up an inequality involving the discriminant and solve it.

判别式也常用于一类题目：“求 \( k \) 的取值范围，使得方程有两个实根”。这类题需要建立关于判别式的不等式并求解。

Remember that for two real roots, the discriminant must be strictly greater than zero. For the equation to touch the x-axis exactly once, the discriminant must equal zero.

请记住：有两个实根时，判别式必须严格大于零；方程与 x 轴恰好相切一次时，判别式必须等于零。

9. Solving Quadratic Equations Graphically | 用图象解二次方程

Sometimes a question will provide the graph of a quadratic function and ask you to solve an equation using the graph. This tests your ability to read information from a graph, which is a key IGCSE skill.

有时题目会给出二次函数的图象，要求你利用图象解方程。这考查的是从图象中读取信息的能力，是 IGCSE 的关键技能之一。

To solve \( ax^2 + bx + c = 0 \) graphically, you simply look for the points where the graph crosses the x-axis. The x-coordinates of these points are the solutions of the equation.

要用图象法解 \( ax^2 + bx + c = 0 \)，只需要找出图象与 x 轴的交点。这些交点的横坐标就是方程的解。

If the equation is not in the standard form, you may need to rearrange it first. For example, to solve \( x^2 – 2x – 3 = 0 \), you could draw the graph of \( y = x^2 – 2x – 3 \) and read off the x-intercepts, which are \( x = -1 \) and \( x = 3 \).

如果方程不是标准形式，你可能需要先移项整理。例如，要解 \( x^2 – 2x – 3 = 0 \)，可以先画出 \( y = x^2 – 2x – 3 \) 的图象，然后读出 x 轴截距，即 \( x = -1 \) 和 \( x = 3 \)。

You may also be asked to solve equations of the form \( x^2 + bx + c = mx + k \) by finding the intersection points of a parabola and a straight line.

题目还可能要求解形如 \( x^2 + bx + c = mx + k \) 的方程，方法是找出抛物线与直线的交点。

10. Transformations of Quadratic Graphs | 二次函数图象的变换

Quadratic graphs can be transformed by translation or reflection. Understanding these transformations helps you sketch graphs quickly and identify equations from given graphs.

二次函数图象可以通过平移或反射进行变换。理解这些变换有助于快速画图，也能帮助你根据给定图象判断方程。

In the vertex form \( y = a(x – h)^2 + k \), the parameter \( h \) translates the graph horizontally, and \( k \) translates it vertically. If \( h > 0 \), the graph shifts to the right; if \( h < 0 \), it shifts to the left. If \( k > 0 \), the graph shifts upward; if \( k < 0 \), it shifts downward.

在顶点式 \( y = a(x – h)^2 + k \) 中，参数 \( h \) 控制水平平移，\( k \) 控制垂直平移。当 \( h > 0 \) 时，图象向右平移；当 \( h < 0 \) 时，向左平移。当 \( k > 0 \) 时，图象向上平移；当 \( k < 0 \) 时，向下平移。

Changing the sign of \( a \) reflects the graph in the x-axis. For example, \( y = -x^2 \) is the reflection of \( y = x^2 \) across the x-axis, turning the minimum point into a maximum point.

改变 \( a \) 的符号会使图象关于 x 轴反射。例如，\( y = -x^2 \) 是 \( y = x^2 \) 关于 x 轴的反射，最小值点变为最大值点。

Multiplying the whole function by a constant \( k \), such as \( y = kx^2 \), stretches or compresses the graph vertically. If \( |k| > 1 \), the graph becomes narrower; if \( 0 < |k| < 1 \), it becomes wider.

将整个函数乘以常数 \( k \)，如 \( y = kx^2 \)，会使图象在垂直方向拉伸或压缩。当 \( |k| > 1 \) 时，图象变得更窄；当 \( 0 < |k| < 1 \) 时，图象变得更宽。

11. Word Problems and Applications | 应用题与实际应用

Quadratic graphs are not just abstract curves; they model many real-world situations. In IGCSE exams, you may encounter problems involving projectile motion, area optimisation, and revenue maximisation.

二次函数图象不仅仅是抽象的曲线，它可以模拟许多现实情境。在 IGCSE 考试中，你可能会遇到涉及抛体运动、面积优化和收益最大化的问题。

In projectile motion problems, the height of an object is often modelled by a quadratic function of time. The maximum height is the y-coordinate of the vertex, and the time at which it reaches the ground is the positive root of the equation.

在抛体运动问题中，物体的高度通常可以表示为时间的二次函数。最大高度就是顶点的 y 坐标，而落地时间就是方程的正根。

In area optimisation problems, you might be asked to find the maximum area of a rectangle given a fixed perimeter. By writing the area as a quadratic function, the maximum area can be found at the vertex.

在面积优化问题中，题目可能要求你在固定周长下求矩形面积的最大值。将面积写成二次函数后，最大值出现在顶点处。

Always identify the variable, write the quadratic function, and then use the vertex formula or completing the square to find the required value. Finally, check whether the answer makes sense in the context of the problem.

解题时，先确定变量，写出二次函数，然后使用顶点公式或配方法求出所需值。最后要检查答案是否符合题目的实际背景。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks in quadratic graph questions due to avoidable mistakes. Here are the most common errors and some practical tips to help you score full marks.

许多学生在二次函数图象题目中因为可以避免的错误而失分。以下是最常见的错误和一些实用技巧，帮助你拿到满分。

Mistake 1: Forgetting that \( a \neq 0 \). If \( a = 0 \), the function is linear, not quadratic.

错误一：忘记 \( a \neq 0 \)。如果 \( a = 0 \)，函数是一次函数，而非二次函数。
Mistake 2: Using the wrong sign when reading the vertex from vertex form. In \( y = a(x – h)^2 + k \), the vertex is \( (h, k) \), so \( y = (x + 2)^2 \) has vertex at \( (-2, 0) \), not \( (2, 0) \).

错误二：从顶点式读取顶点时用错符号。在 \( y = a(x – h)^2 + k \) 中，顶点为 \( (h, k) \)，所以 \( y = (x + 2)^2 \) 的顶点是 \( (-2, 0) \)，而不是 \( (2, 0) \)。
Mistake 3: Forgetting that the y-intercept is \( c \) only when the equation is written in expanded form.

错误三：忘记只有方程写成展开形式时，y 轴截距才是 \( c \)。
Mistake 4: Drawing the graph too steep or too flat. Use a table of values for accuracy when required.

错误四：图象画得太陡或太平。需要时用数值表格来保证准确。
Tip 1: Always show the axis of symmetry in your sketch.

技巧一：草图中始终画出对称轴。
Tip 2: When the discriminant is negative, state clearly that there are no real roots.

技巧二：当判别式为负时，清楚说明没有实数根。
Tip 3: Check your vertex by substituting it back into the original equation.

技巧三：将顶点坐标代回原方程进行验证。

Quadratic functions are a fundamental part of IGCSE Mathematics. Once you master the general form, roots, vertex, and transformations, you will find that most quadratic graph questions follow a very similar pattern. Practise sketching graphs by hand regularly, and you will build both speed and confidence.

二次函数是 IGCSE 数学的基础内容。一旦你掌握了一般形式、根、顶点和变换，你会发现大多数二次函数图象题目都非常相似。经常练习手绘草图，你的速度和信心都会不断提升。

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Graphs of Quadratic Functions | 二次函数图象