📚 IB Mathematics: Function Graph Transformations: f(ax+b) and [f(x)]² | IB数学：函数图像变换：f(ax+b)与[f(x)]²
Function graph transformations are a cornerstone of IB Mathematics. Two important types are composite linear transformations represented by f(ax+b), and the power transformation [f(x)]². Understanding these helps you sketch graphs quickly and interpret relationships between functions.
函数图像变换是IB数学的核心内容之一。两个重要类型是 f(ax+b) 所代表的线性复合变换，以及 [f(x)]² 的幂变换。理解这些变换能帮助你快速作图，并解释函数之间的关系。
1. Understanding f(ax+b): Re-expression | 理解 f(ax+b)：改写形式
To analyse f(ax+b), rewrite it as f(a(x + b/a)). The graph of y = f(ax+b) is obtained from y = f(x) by a horizontal stretch/compression by factor 1/|a|, followed by a horizontal translation by -b/a. If a is negative, there is also a reflection in the y-axis.
要分析 f(ax+b)，先将其改写为 f(a(x + b/a))。y = f(ax+b) 的图像可由 y = f(x) 先作水平伸缩（伸缩因子为 1/|a|），再作水平平移（平移量为 -b/a）得到。若 a 为负，还需关于 y 轴作反射。
For example, f(2x+1) = f(2(x+½)). Start with y = f(x). Replace x by 2x to obtain a horizontal compression by factor ½. Then replace x by x+½, which shifts the compressed graph left by ½.
例如，f(2x+1) = f(2(x+½))。从 y = f(x) 开始，将 x 替换为 2x 得到水平压缩（因子为 ½）。再将 x 替换为 x+½，即把压缩后的图像向左平移 ½。
2. Key Point Mapping for f(ax+b) | f(ax+b) 的关键点映射
For a point (p, q) on y = f(x), the corresponding point on y = f(ax+b) is ( (p – b)/a , q ), provided a ≠ 0. This formula combines all transformations into one step, and is especially useful for sketching.
对于 y = f(x) 上的点 (p, q)，在 y = f(ax+b) 上对应的点为 ( (p – b)/a , q )，其中 a ≠ 0。这个公式将全部变换合并为一步，特别适合作图时使用。
Consider f(x) = x². The table below shows how selected points transform under f(2x+1).
以 f(x) = x² 为例。下表展示了若干点在 f(2x+1) 下的变换。
| Point on f(x) | Point on f(2x+1) |
| (-1, 1) | ((-1-1)/2, 1) = (-1, 1) |
| (0, 0) | ((0-1)/2, 0) = (-½, 0) |
| (1, 1) | ((1-1)/2, 1) = (0, 1) |
| (2, 4) | ((2-1)/2, 4) = (½, 4) |
| (4, 16) | ((4-1)/2, 16) = (3/2, 16) |
3. Common Pitfalls with f(ax+b) | f(ax+b) 的常见误区
One common mistake is to stretch first and then shift by b. This produces f(a(x+b)) = f(ax+ab), not f(ax+b). The correct shift after stretching is by b/a, not by b.
一个常见错误是先作伸缩，然后再平移 b。这会得到 f(a(x+b)) = f(ax+ab)，而不是 f(ax+b)。在伸缩之后，正确的平移量应为 b/a，而不是 b。
Another mistake is forgetting to factor out a when b ≠ 0. Always rewrite as f(a(x + b/a)) before applying transformations. Also note that if a = -1, the reflection in the y-axis must be applied, and the shift formula still works.
另一个错误是在 b ≠ 0 时忘记提取因子 a。在应用变换前，务必将表达式改写为 f(a(x + b/a))。另外，若 a = -1，必须应用关于 y 轴的反射，平移公式仍然适用。
4. Introduction to [f(x)]² | [f(x)]² 简介
The function y = [f(x)]² is defined as (f(x))². Its domain is the same as the domain of f, because every real number can be squared. The range, however, changes: only non-negative values appear, since a square is never negative.
函数 y = [f(x)]² 定义为 (f(x))²。其定义域与 f 的定义域相同，因为任何实数都可以平方。但值域发生了变化：由于平方永远非负，所以只出现非负值。
For example, if f(x) = √x for x ≥ 0, then [f(x)]² = x, but only for x ≥ 0. In this case the graph becomes a straight ray.
例如，若 f(x) = √x（x ≥ 0），则 [f(x)]² = x，但仅在 x ≥ 0 时成立。此时图像变成一条射线。
5. Sign Analysis: Zeros and Positivity | 符号分析：零点与正值
The graph of y = [f(x)]² lies on or above the x-axis. The zeros of [f(x)]² are exactly the zeros of f(x). At these points the graph touches the x-axis and does not cross it, because the function value is non-negative on both sides.
y = [f(x)]² 的图像位于 x 轴上方或与 x 轴相切。[f(x)]² 的零点恰好是 f(x) 的零点。在这些点处，图像接触 x 轴但不穿越，因为函数值在两侧均为非负。
If f(x) > 0 for some interval, then [f(x)]² is also positive; if f(x) < 0, then [f(x)]² is again positive. Therefore all negative parts of f(x) are reflected above the x-axis and then stretched or compressed vertically.
若在某个区间上 f(x) > 0，则 [f(x)]² 也为正；若 f(x) < 0，则 [f(x)]² 仍然为正。因此 f(x) 的负值部分会被翻折到 x 轴上方，再进行竖直方向的伸缩或压缩。
6. Magnitude Effects: |f(x)| vs [f(x)]² | 数值效应：|f(x)| 与 [f(x)]²
The relationship between the graphs of y = f(x) and y = [f(x)]² depends heavily on the magnitude of f(x).
y = f(x) 与 y = [f(x)]² 的图像关系很大程度上取决于 f(x) 的绝对值大小。
-
Where |f(x)| > 1, the squared value is larger: |[f(x)]²| > |f(x)|. The graph is vertically stretched.
当 |f(x)| > 1 时，平方后的数值更大：|[f(x)]²| > |f(x)|。图像被竖直拉伸。
-
Where |f(x)| < 1, the squared value is smaller: |[f(x)]²| < |f(x)|. The graph is pulled toward the x-axis.
当 |f(x)| < 1 时，平方后的数值更小：|[f(x)]²| < |f(x)|。图像被拉近 x 轴。
-
Where |f(x)| = 1, the point is fixed on the horizontal line y = 1. Note that both f(x) = 1 and f(x) = -1 map to y = 1.
当 |f(x)| = 1 时，点固定在水平线 y = 1 上。注意 f(x) = 1 和 f(x) = -1 都映射到 y = 1。
This magnitude analysis is the most practical tool for sketching [f(x)]² quickly.
这种数值大小分析是快速绘制 [f(x)]² 图像的最实用工具。
7. Effects on Extrema and Shape | 对极值与形状的影响
Squaring can turn a local maximum into a local minimum, depending on the sign of f(x) near the extremum.
平方变换可以将局部极大值变成局部极小值，具体取决于极值点附近 f(x) 的符号。
If f has a local maximum at x = a with value M > 0, then near a the values of f(x) are ≤ M, so their squares are ≤ M². Thus (a, M²) is a local maximum. If M < 0, nearby values are more negative (e.g., -3 is less than -2), so their squares become larger than M². The same point becomes a local minimum of [f(x)]².
若 f 在 x = a 处取得局部极大值 M > 0，则在 a 附近 f(x) ≤ M，因此其平方 ≤ M²，于是 (a, M²) 是局部极大值。若 M < 0，附近的值会更负（例如 -3 小于 -2），于是平方后大于 M²，同一个点变成 [f(x)]² 的局部极小值。
Similarly, a negative local minimum (a “valley” below the x-axis) will produce a local maximum after squaring, because nearby values are less negative than the minimum, yielding smaller squares.
类似地，负的局部极小值（x 轴下方的“谷”）在平方后会变成局部极大值，因为附近的值不如极小值那么负，平方反而更小。
8. Worked Example: f(x) = x+1 vs [f(x)]² |
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply