Graphs 8: Transformations and Graphical Solutions | 图像8：图像变换与图解求解

📚 Graphs 8: Transformations and Graphical Solutions | 图像8：图像变换与图解求解

Graphs are one of the most visual topics in IGCSE Mathematics. In this lesson, we complete our study of the Graphs series by investigating how known curves can be transformed by reflection, translation and stretch, and how a carefully drawn graph can be used to solve equations that are hard to solve by algebra alone.

图像是IGCSE数学中最具视觉性的主题之一。在本课中，我们将完成“图像”系列的最后一讲：研究已知曲线如何通过反射、平移和伸缩进行变换，并学习如何借助精心绘制的图像来求解那些仅靠代数难以处理的方程。

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1. Recap: Quadratic Graphs | 回顾：二次函数图像

A quadratic function has the form y = ax² + bx + c, where a is not zero. Its graph is a parabola. If a > 0, the curve is U-shaped and has a minimum point; if a < 0, the curve is n-shaped and has a maximum point.

二次函数的一般形式为y = ax² + bx + c，其中a不等于零。它的图像为抛物线：若a > 0，曲线呈“U”形，有最小值点；若a < 0，曲线呈“n”形，有最大值点。

To sketch a quadratic, find the roots by solving ax² + bx + c = 0, find the y-intercept by substituting x = 0, and find the turning point by completing the square or using symmetry.

绘制二次函数图像时，先通过解ax² + bx + c = 0求出根，再令x = 0求y截距，最后通过配方或利用对称性确定顶点。

For example, y = x² − 4x + 3 factorises as (x − 1)(x − 3). The roots are x = 1 and x = 3, the y-intercept is 3, and completing the square gives y = (x − 2)² − 1, so the minimum point is (2, −1).

例如，y = x² − 4x + 3可因式分解为(x − 1)(x − 3)。两根为x = 1与x = 3，y截距为3，配方得y = (x − 2)² − 1，故最小值点为(2, −1)。

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2. Cubic and Reciprocal Graphs | 三次函数与反比例函数图像

A cubic function has the form y = ax³ + bx² + cx + d. When a is positive, the curve rises from bottom-left to top-right; when a is negative, it falls from top-left to bottom-right. A cubic can cross the x-axis at up to three points.

三次函数的一般形式为y = ax³ + bx² + cx + d。当a为正时，曲线从左下方向右上方延伸；当a为负时，曲线从左上方向右下方延伸。三次函数的图像最多可与x轴有三个交点。

For instance, y = x³ − 4x = x(x − 2)(x + 2) crosses the x-axis at x = −2, 0 and 2. Between the roots, the curve rises and falls smoothly, producing one or two turning points.

例如，y = x³ − 4x = x(x − 2)(x + 2)在x = −2、0和2处穿过x轴。在根之间，曲线平滑地上升和下降，形成一个或两个转向点。

The reciprocal graph y = k/x has two separate branches and two asymptotes: the x-axis and the y-axis. If k > 0, the branches lie in the first and third quadrants; if k < 0, they lie in the second and fourth quadrants.

反比例函数y = k/x的图像有两条分离的分支以及两条渐近线：x轴和y轴。若k > 0，两条分支位于第一和第三象限；若k < 0，则位于第二和第四象限。

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3. Exponential Graphs | 指数函数图像

An exponential function has the form y = aˣ, where a > 0 and a ≠ 1. The graph always passes through (0, 1), because any nonzero number raised to the power 0 equals 1.

指数函数的一般形式为y = aˣ，其中a > 0且a ≠ 1。图像始终经过点(0, 1)，因为任何非零数的0次幂都等于1。

If a > 1, the curve shows exponential growth: it rises steeply as x increases and approaches the x-axis as x decreases. If 0 < a < 1, the curve shows exponential decay: it falls towards zero as x increases.

若a > 1，曲线呈指数增长：随x增大而急剧上升，随x减小而趋近x轴。若0 < a < 1，曲线呈指数衰减：随x增大而趋近于零。

The x-axis is a horizontal asymptote, so the curve never actually touches it. For example, y = 2ˣ passes through (0, 1) and (1, 2), and y = 2⁻ˣ is the reflection of y = 2ˣ in the y-axis.

x轴是水平渐近线，因此曲线永远不会真正触碰到它。例如，y = 2ˣ经过点(0, 1)和(1, 2)，而y = 2⁻ˣ是y = 2ˣ关于y轴的反射。

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4. Trigonometric Graphs | 三角函数图像

Trigonometric graphs are periodic, meaning that they repeat their shape at regular intervals. The graph of y = sin θ has amplitude 1 and period 360°, with values always between −1 and 1.

三角函数图像是周期性的，即其形状按固定间隔重复。y = sin θ的图像振幅为1，周期为360°，函数值始终在−1与1之间。

The graph of y = cos θ has the same amplitude and period as the sine graph, but it starts at (0, 1). The cosine graph is the sine graph shifted 90° to the left.

y = cos θ的图像与正弦图像具有相同的振幅和周期，但它从点(0, 1)出发。余弦图像是正弦图像向左平移90°得到的。

The graph of y = tan θ is different: it has period 180° and has vertical asymptotes at θ = 90°, 270°, 450° and so on. The curve rises steeply and never crosses these asymptotes.

y = tan θ的图像则不同：它的周期为180°，在θ = 90°、270°、450°等处有垂直渐近线。曲线陡峭上升，绝不会越过这些渐近线。

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5. Reflection Transformations | 反射变换

The transformation y = −f(x) reflects the graph of y = f(x) in the x-axis. Every point (x, y) moves to (x, −y), so the curve is flipped vertically.

变换y = −f(x)将y = f(x)的图像关于x轴反射。每个点(x, y)移动至(x, −y)，即曲线上下翻转。

The transformation y = f(−x) reflects the graph in the y-axis. Every point (x, y) moves to (−x, y), so the curve is flipped horizontally.

变换y = f(−x)将图像关于y轴反射。每个点(x, y)移动至(−x, y)，即曲线左右翻转。

For example

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