A-Level数学考点-函数图像变换Transformation

函数是A-level数学考试中常考话题，下面是重庆新航道alevel培训老师，给大家解析的alevel数学-函数图像变换，希望能帮到大家。

What are graph transformations?

When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations. There are three types of transformations: translation，stretch and reflection.

什么是图像变换?当以某种方式改变一个函数时，对函数图像的影响就被称为几何变换。变换有三种：平移、拉伸和对称。

1.Translation平移

Adding or subtracting a constant‘inside’ or‘outside’ the function translates a graph horizontally or vertically. With a translation the shape and size of the graph remain unchanged.

在函数的 “内部 ”或 “外部 ”加或减一个常数，函数图像会水平或垂直移动, 而形状、大小都保持不变。

① horizontal translation(左右平移)

f(x)→f(x±a)，a>0

The graph moves left for positive values of a and right for negative values of a，

the y-coordinates stay the same【左加右减】

坐标变化为：x坐标减加a，y坐标不变

② vertical translation (上下平移)

f(x)→f(x)±a，a>0

The graph moves up for positive values of a and down for negative values of a，the x-coordinates stay the same【上加下减】

坐标变化为：y坐标加减a，x坐标不变

2.Stretch拉伸

Multiplying by a constant‘inside’ or‘outside’ the function stretches the graph horizontally or vertically. With a stretch all the points on the graph are moved towards or away from either the x or the y axis by a constant scale factor.

在函数的 “内部 ”或 “外部 ”乘以一个常数，可以水平或垂直拉伸图像。在拉伸过程中，图像上的所有点都会以一个恒定的比例靠近x轴y轴或远离x轴y轴。

① horizontal stretch (水平拉伸)

l f(x)→f(ax),(a>0)

The x coordinates are multiplied by 1/a; y coordinates of points stay the same

坐标变化：x坐标×1/a, y坐标不变

② vertical stretch (垂直拉伸)

f(x)→af(x),(a>0)

The y coordinates are multiplied by a; x coordinates of points stay the same

坐标变化：y坐标×a , x坐标不变

3. Reflections对称

Multiplying by ‘-1’‘inside’ or‘outside’ the function reflects the graph in the y-axis or x-axis. 在函数的 “内部 ”或 “外部 ”乘以“-1”，图像会关于y轴或x轴对称。

① reflection in the y-axis(关于y轴对称)

f(x)→f(-x)

The y coordinates of points stay the same; x coordinates have their signs changed (positive to negative, negative to positive)

坐标变化：x坐标变为相反数，y坐标不变

②reflection in the x-axis(关于x轴对称)

f(x)→-f(x)

The x coordinates of points stay the same; y coordinates have their signs changed (positive to negative, negative to positive)

坐标变化：y坐标变为相反数，x坐标不变

4.Brief summary：

①translation

f(x)→f(x±a)：左加右减，x坐标减加a，y坐标不变

f(x)→f(x)±a：上加下减，y坐标加减a，x坐标不变

②stretch

f(x)→f(ax)：x坐标×1/a, y坐标不变

f(x)→af(x)：y坐标×a , x坐标不变，

③reflection

f(x)→f(-x)：关于y轴对称，x坐标变为相反数，y坐标不变

f(x)→-f(x)：关于x轴对称，y坐标变为相反数，x坐标不变

Exam Tips:

When you sketch the transformed graph:在绘制变换后的函数图像时:

be sure to indicate the new coordinates of any points that are marked on the original graph;

需标出原图像上标注的点的新坐标;

be sure to indicate the specific coordinates of points as required: intersections with the coordinate axes, vertices or the turning points;

需标出题目要求的点的坐标：比如与坐标轴的交点、顶点或转折点;

don't forget to sketch the new asymptotes of the transformed graph as well if the original graph has asymptotes.

原图像如有渐近线，需在新图像中画出变换后的渐近线并标出渐近线方程。

Worked example:

详解：

(i)y=f(x)-2是在f(x)后面减2，说明是上下平移;

上加下减，说明要向下平移2个单位;

坐标变化：新图像y坐标减2，x坐标不变：P(-1,0)→(-1，-2);Q(3/2,2)→(3/2,0);

渐近线变化：有渐近线y=1，图像向下平移2个单位, 渐近线也要向下平移2个单位：

y=1→y=-1。

所以新图像为：(新图像的P点Q点渐近线都要标出)

(ii)y=f(-x)是在x前面乘了-1，说明图像关于y轴对称;

坐标变化：新图像x坐标变为相反数，y坐标不变：P(-1,0)→(1,0);Q(3/2,2)→(-3/2,2);

渐近线变化：有渐近线y=1, 但图像关于y轴对称(y坐标不变)，所以渐近线不变：y=1。

所以新图像为：(新图像的P点Q点渐近线都要标出)。