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)。
