📚 Graphs of Functions and Turning Points | 函数图像与转折点
The study of function graphs is central to IGCSE Mathematics because it links algebra, coordinate geometry, and real-world modelling. Understanding how to sketch, interpret, and transform graphs helps you solve equations visually and analyse turning points, intercepts, and asymptotic behaviour with confidence.
函数图像的学习是 IGCSE 数学的核心内容，因为它将代数、坐标几何和现实建模联系在一起。理解如何绘制、解读和变换图像，有助于你直观地解方程，并自信地分析转折点、截距和渐近行为。
1. Coordinates and the Cartesian Plane | 坐标与笛卡尔平面
The Cartesian plane is formed by two number lines crossing at right angles: the horizontal x-axis and the vertical y-axis. Every point is written as an ordered pair (x, y), where x gives the horizontal position and y gives the vertical position. Understanding coordinates is the foundation for plotting and interpreting graphs.
笛卡尔平面由两条互相垂直的数轴组成：水平的 x 轴和竖直的 y 轴。每个点都写成有序数对 (x, y)，其中 x 表示水平位置，y 表示竖直位置。理解坐标是绘制和解读函数图像的基础。
The four quadrants are labelled anticlockwise, starting from the top right as the first quadrant, where both x > 0 and y > 0. In the second quadrant, x < 0 and y > 0; in the third quadrant, both are negative; and in the fourth quadrant, x > 0 while y < 0.
四个象限按逆时针方向标记，从右上方的第一象限开始，第一象限中 x > 0 且 y > 0。第二象限中 x < 0 且 y > 0；第三象限中两者都为负；第四象限中 x > 0 而 y < 0。
2. Linear Functions and Gradient | 线性函数与斜率
asures the steepness of the line and is calculated as m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁).
线性函数的一般形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率度量直线的倾斜程度，计算公式为 m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)。
y = mx + c
es have equal gradients, while two perpendicular lines satisfy the condition m₁ × m₂ = −1.
如果 m > 0，直线从左到右向上倾斜；如果 m < 0，直线从左到右向下倾斜。两条平行线的斜率相等，而两条垂直线的斜率满足条件 m₁ × m₂ = −1。
3. Quadratic Functions and Parabolas | 二次函数与抛物线
< 0. The y-intercept is c, and the x-intercepts are found by solving ax² + bx + c = 0.
二次函数的形式为 y = ax² + bx + c，其中 a ≠ 0。它的图像是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。y 轴截距为 c，x 轴截距通过解方程 ax² + bx + c = 0 求得。
y = ax² + bx + c
lue is essential for locating the turning point and for sketching the parabola accurately.
对称轴穿过顶点。对于标准形式的二次函数，顶点的 x 坐标为 x = −b ÷ (2a)。这个值是确定转折点位置和准确绘制抛物线的关键。
4. Completing the Square and Vertex Form | 配方法与顶点式
² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the turning point and the direction of opening.
配方法将 y = ax² + bx + c 转化为顶点式 y = a(x − h)² + k，其中 (h, k) 是抛物线的顶点。这种形式便于确定转折点和开口方向。
The axis of symmetry is x = 3, and because a = 1 is positive, the parabola opens upwards.
例如，表达式 y = x² − 6x + 5 可以改写为 y = (x − 3)² − 4，因此顶点为 (3, −4)。对称轴为 x = 3，并且由于 a = 1 为正，抛物线开口向上。
y = x² − 6x + 5 → y = (x − 3)² − 4
5. Finding Turning Points | 求转折点
ing if you study further mathematics. In IGCSE, completing the square is often the most reliable method.
对于二次函数，转折点就是顶点。当 a > 0 时为最小值点，当 a < 0 时为最大值点。x 坐标可以用 x = −b ÷ (2a) 求出，如果你学习进阶数学也可以通过求导得到。在 IGCSE 中，配方法通常是最可靠的方法。
ing point is (1 + 5) ÷ 2 = 3. Substituting x = 3 into the equation gives the y-coordinate.
你也可以利用对称性：转折点正好位于两个 x 轴截距的中点。例如，如果根为 x = 1 和 x = 5，则转折点的 x 坐标为 (1 + 5) ÷ 2 = 3。将 x = 3 代入方程即可得到 y 坐标。
6. Sketching Cubic and Reciprocal Graphs | 绘制三次函数与倒数函数图像
ctions of the form y = k ÷ x have two separate branches and asymptotes at x = 0 and y = 0.
三次函数的形式为 y = ax³ + bx² + cx + d，其图像呈 S 形。a 的符号决定图像向右延伸时是上升还是下降。形如 y = k ÷ x 的倒数函数有两条独立的分支，并且在 x = 0 和 y = 0 处有渐近线。
| Function type | Basic shape | Key features |
|---|---|---|
| Linear y = mx + c | Straight line | Gradient m, y-intercept c |
| Quadratic y = ax² + bx + c | Parabola | Vertex, axis of symmetry, intercepts |
| Cubic y = ax³ + bx² + cx + d | S-shaped curve | Up to two turning points, intercepts |
| Reciprocal y = k ÷ x | Hyperbola | Asymptotes x = 0, y = 0 |
When sketching any of these graphs, identify the intercepts, turning points, and asymptotic behaviour before drawing the curve. This helps you produce a clear, correctly shaped graph even without plotting many points.
在绘制这些图像时，先确定截距、转折点和渐近行为，再画出曲线。这有助于你在不需要描出大量点的情况下，画出清晰且形状正确的图像。
7. Intersections and Solving Equations Graphically | 图像交点与解方程
ot y = x² and y = x + 2 on the same axes and read the x-values of the intersection points.
方程 f(x) = g(x) 的解对应于图像 y = f(x) 与 y = g(x) 相交处的 x 坐标。例如，要求解 x² = x + 2，你可以在同一坐标系中画出 y = x² 和 y = x + 2，然后读取交点的 x 值。
e the solutions. This method is especially useful for equations that are difficult to rearrange algebraically, and it appears frequently in IGCSE graph questions.
在上面的例子中，图像在 x = −1 和 x = 2 处相交，因此这些就是方程的解。对于难以用代数方法重新整理的方程，这种方法尤其有用，并且在 IGCSE 图像题中经常出现。
x² = x + 2 → x = −1 or x = 2
8. Transformations of Graphs | 图像变换
orizontally; y = −f(x) reflects it in the x-axis; and y = f(−x) reflects it in the y-axis.
图像可以通过平移、反射、伸缩和旋转进行变换。在 IGCSE 中，主要变换包括：y = f(x) + a 将图像竖直平移；y = f(x + a) 将其水平平移；y = −f(x) 使其关于 x 轴反射；y = f(−x) 使其关于 y 轴反射。
s to the right and 3 units upward. The vertex moves from (0, 0) to (2, 3), but the shape of the parabola remains unchanged.
例如，y = (x − 2)² + 3 的图像就是 y = x² 的图像向右平移 2 个单位、向上平移 3 个单位。顶点从 (0, 0) 移动到 (2, 3)，但抛物线的形状保持不变。
y = (x − 2)² + 3 → vertex (2, 3)
9. Real-World Applications | 实际应用
Graphs of functions model real-world relationships such as projectile motion, profit and cost, speed-time graphs, and growth or decay. A quadratic function can model the height of a ball thrown upwards, while a reciprocal function can model inverse proportion between two quantities.
函数图像可以用来模拟现实世界中的关系，例如抛体运动、利润与成本、速度-时间图以及增长或衰减。二次函数可以模拟抛出的球的高度，而倒数函数可以模拟两个量之间的反比例关系。
ll in metres after t seconds, the maximum height occurs at t = −20 ÷ (2 × −5) = 2 seconds.
解读转折点通常是关键任务：最大高度或最大利润就是顶点的 y 坐标。例如，如果 h = −5t² + 20t + 1 表示球在 t 秒后的高度（米），那么最大高度出现在 t = −20 ÷ (2 × −5) = 2 秒时。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always label the axes and key points when drawing graphs. Show your working for finding intercepts and turning points, and use a smooth curve rather than straight line segments for quadratic and cubic graphs. Read the question carefully to see whether an exact or approximate solution is required.
画图时始终标注坐标轴和关键点。展示求截距和转折点的过程，并且对于二次函数和三次函数图像要用光滑曲线，而不是直线段。仔细审题，明确要求的是精确解还是近似解。
cepts, and misreading the scale on a graph. Always check your sketch against the sign of the leading coefficient: a positive a gives an upward-opening parabola, while a negative a gives a downward-opening parabola.
常见错误包括二次公式中的符号错误、求 x 轴截距时忘记令 y = 0，以及误读图像上的刻度。始终根据首项系数的符号检查你的草图：a 为正时抛物线开口向上，a 为负时抛物线开口向下。
Published by TutorHao | Maths 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