📚 Function Stretches | 函数拉伸变换
In IB Mathematics, transforming graphs is a fundamental skill, and stretches are one of the key transformations that change the shape of a function's graph without altering its basic identity. Understanding vertical and horizontal stretches—and how they differ from compressions—enables you to manipulate and analyse functions effectively, whether you are working with parabolas, trigonometric curves, or exponential graphs.
在 IB 数学中，图形变换是一项基本技能，而拉伸是改变函数图形形状但不改变其基本特征的关键变换之一。理解垂直拉伸与水平拉伸（以及它们与压缩的区别），能让你有效地操作和分析函数，无论是处理抛物线、三角曲线还是指数图像。
1. What Are Function Stretches? | 什么是函数拉伸？
A stretch is a transformation that alters the size of a graph in either the vertical or horizontal direction. Unlike translations (shifts) that slide the graph, stretches pull or squeeze it away from or towards an axis by a constant scale factor. The overall shape remains similar—angles at critical points are preserved, but distances change proportionally.
拉伸是一种变换，会改变图像在垂直或水平方向上的大小。与滑动图像的平移不同，拉伸通过一个恒定的比例因子将图像拉离或压向坐标轴。整体形状保持相似——关键点处的角度不变，但距离按比例改变。
Stretches are described by multiplicative constants in the function' tal change. The values of a and b determine whether the transformation is a stretch (expansion) or a compression.
拉伸由函数表达式中的乘法常数描述：函数外部的系数 y = a f(x) 产生垂直拉伸；括号内自变量的系数 y = f(b x) 产生水平变化。a 和 b 的值决定了变换是拉伸（扩大）还是压缩。
2. Vertical Stretches: y = a f(x) | 垂直拉伸：y = a f(x)
sitive real number. For any point (x, y) on the original graph, its image after the transformation is (x, a y).
垂直拉伸将原函数的每个 y 坐标乘以常数 a。该变换写作 y = a f(x)，其中 a 是一个正实数。对于原图像上的任意点 (x, y)，其变换后的像点为 (x, a y)。
If a > 1, the graph moves away from the x‑axis, so the shape appears taller—this is a vertical stretch. If 0 < a < 1, the graph is pushed towards the x‑axis, making it look flatter; this is actually a vertical compression, though the same formula applies. When a is negative, an additional reflection in the x‑axis occurs, but the magnitude |a| still governs the stretch or compression.
如果 a > 1，图像远离 x 轴，形状显得更高——这是垂直拉伸。如果 0 < a < 1，图像被推向 x 轴，看起来更扁平；这实际上是垂直压缩，但公式仍然适用。当 a 为负时，还会发生关于 x 轴的反射，但 |a| 的大小仍然控制着拉伸或压缩。
abola by a factor of 3, while y = ½ f(x) = ½ x² compresses it vertically by a factor of 2.
例如，若 f(x) = x²，那么 y = 3 f(x) = 3x² 将抛物线垂直拉伸为原来的 3 倍，而 y = ½ f(x) = ½ x² 将抛物线垂直压缩为原来的一半（因子为 2）。
3. Effects of Vertical Stretches on Key Points | 垂直拉伸对关键点的影响
Because vertical stretches multiply y‑coordinates, they mainly affect the range of the function and the locations of maxima, minima, and intercepts on the y‑axis. The x‑intercepts (roots) remain unchanged if a ≠ 0, because multiplying zero by a still gives zero. Turning points move vertically: a maximum at (p, q) becomes (p, a q).
由于垂直拉伸乘以 y 坐标，它们主要影响函数的值域以及最大值、最小值和 y 轴截距的位置。若 a ≠ 0，x 截距（根）保持不变，因为 0 乘以 a 仍为 0。极值点垂直移动：最大值点 (p, q) 变为 (p, a q)。
om 1 to 2, so the graph oscillates between -2 and 2 instead of -1 and 1. The period stays the same. Thus vertical stretches alter the output scale without changing the input cycle.
对于三角函数，如 y = 2 sin(x)，振幅从 1 加倍至 2，图像在 -2 与 2 之间振荡，而不是 -1 与 1。周期保持不变。因此，垂直拉伸改变输出的尺度，而不改变输入的循环。
4. Horizontal Stretches: y = f(b x) | 水平拉伸：y = f(b x)
n) transforms the function to y = f(b x), where b is a positive constant. This time, the x‑ : for every point (x, y) on the original graph, the transformed point is (x / b, y). The y‑coordinates do not change directly, but the graph's horizontal spread alters.
水平拉伸（或压缩）将函数变换为 y = f(b x)，其中 b 是一个正常数。此时，x 坐标被缩放：原图像上的每一点 (x, y)，变换后的点为 (x / b, y)。y 坐标不直接改变，但图像的水平延展度发生变化。
might suggest at first glance. If b > 1, the graph is squeezed horizontally towards the y‑axis by a factor of 1/b— horizontal compression. If 0 < b < 1, the graph is stretched horizontally away from the y‑axis— r of 2, while y = f(x/2), which is y = f(½ x), stretches it horizontally by a factor of 2.
其效果与数字 b 的直观感觉相反。如果 b > 1，图像在水平方向上被压缩，向 y 轴靠拢，压缩因子为 1/b——这是水平压缩。如果 0 < b < 1，图像在水平方向上被拉伸，远离 y 轴——这是水平拉伸。例如，y = f(2x) 将图像水平压缩为原来的 1/2，而 y = f(x/2)（即 y = f(½ x)）将图像水平拉伸为原来的 2 倍。
5. Horizontal Stretch Effects on Coordinates and Period | 水平拉伸对坐标和周期的影响
Because x‑coordinates are divided by b, the positions of vertical features change: the y‑intercept remains the same (when x = 0, 0/b = 0), but x‑ ) has period 2π; sin(2x) has period π (compression), and sin(½ x) has period 4π (stretch).
由于 x 坐标除以 b，垂直特征的位置发生变化：y 截距保持不变（当 x = 0 时，0/b = 0），但 x 截距移动至 (根值 / b, 0)。对于周期函数，周期变为原周期 ÷ b。例如 sin(x) 的周期为 2π；sin(2x) 的周期为 π（压缩），而 sin(½ x) 的周期为 4π（拉伸）。
= b now occurs at x = 1, so the graph speeds up or slows down horizontally depending on b.
这种行为可以这样记忆：对 x 应用 b 的倒数。一个有用的检验是：将 x = 1 代入 f(b x)，原本在 x = b 处出现的输出现在在 x = 1 处出现，因此图像根据 b 在水平方向上加速或减慢。
6. Visualising the Difference Between Stretch and Compression | 直观区分拉伸与压缩
ph behaves relative to the axis. For vertical transformations y = a f(x): larger a (a > 1) ⇢ taller graph ⇢ stretch; smaller a (0 < a < 1) ⇢ flatter graph ⇢ compression.
为了避免混淆，可以将乘数的大小与图形相对坐标轴的行为联系起来。对于垂直变换 y = a f(x)：较大的 a (a > 1) ⇢ 更高的图形 ⇢ 拉伸；较小的 a (0 < a < 1) ⇢ 更扁平的图形 ⇢ 压缩。
For horizontal transformations y = f(b x): larger b (b > 1) ⇢ narrower graph ⇢ compression; smaller b (0 < b < 1) ⇢ wider graph ⇢ stretch. A common mnemonic is “big b narrow, small b wide.” Remember that b comes into the argument, so it affects the spread of x‑values inversely.
对于水平变换 y = f(b x)：较大的 b (b > 1) ⇢ 更窄的图形 ⇢ 压缩；较小的 b (0 < b < 1) ⇢ 更宽的图形 ⇢ 拉伸。一个常见的记忆法是“大 b 窄，小 b 宽”。记住 b 位于自变量中，因此它反向影响 x 值的延展。
7. Examples with Quadratic and Absolute Value Functions | 二次函数和绝对值函数的实例
the parabola horizontally by factor 3, we write y = (x/3)² = x²/9, making it much flatter.
考虑基本抛物线 y = x²。应用因子为 3 的垂直拉伸得到 y = 3x²；顶点仍在 (0,0)，但两臂上升得更陡。使用 y = (2x)² = 4x² 进行因子为 2 的水平压缩也会使两臂变陡，但方式不同。要将抛物线水平拉伸为原来的 3 倍，可写为 y = (x/3)² = x²/9，这样图形会平坦得多。
For the absolute value function y = |x|, a vertical stretch y = 2|x| makes the V‑shape steeper. A horizontal stretch y = |0.5x| = |x/2| makes the V wider. Notice that y = |2x| compresses the V horizontally, producing the same steeper V as y = 2|x|? In this specific case, since |2x| = 2|x| and horizontal stretches can sometimes coincide, but this is not true for most functions.
对于绝对值函数 y = |x|，垂直拉伸 y = 2|x| 使 V 形更陡。水平拉伸 y = |0.5x| = |x/2| 使 V 形更宽。注意 y = |2x| 将 V 形水平压缩，产生与 y = 2|x| 同样陡峭的 V 形。在这个特例中，由于 |2x| = 2|x|，垂直与水平拉伸有时会一致，但对大多数函数并非如此。
8. Combining Stretches with Translations | 拉伸与平移的组合
by k. Horizontal scaling should be applied to the input after the shift is taken into account. If the function is written as y = a f(b x - c), be careful—r b from both terms inside the bracket: f(b(x - c/b)). The horizontal shift is c/b, not c.
当一个函数包含多种变换时，顺序很重要，因为拉伸会影响已有的平移。一般形式 y = a f(b(x - h)) + k 表示：水平平移 h，水平拉伸/压缩因子 1/b，垂直拉伸/压缩 a，垂直平移 k。水平缩放应在考虑平移之后应用于输入。如果函数写作 y = a f(b x - c)，要小心——将括号内的两项提取 b：f(b(x - c/b))。水平平移量是 c/b，而不是 c。
etched first to 3x² and then shifted, we would get 3(x - 2)² as well, because vertical stretch and horizontal translation commute. However, horizontal stretch and horizontal translation do not commute; always factor the horizontal coefficient correctly.
例如，假设我们要对 f(x) = x² 先向右平移 2 个单位，然后垂直拉伸为原来的 3 倍。正确的表达式为 3 f(x - 2) = 3(x - 2)²。如果先拉伸为 3x² 再平移，我们仍然得到 3(x - 2)²，因为垂直拉伸与水平平移可交换。然而，水平拉伸与水平平移不可交换；务必正确提取水平系数。
9. Identifying Stretch Factors from a Graph | 从图像识别拉伸因子
(not on an axis), find the corresponding point on the original function, and compute a = yimage / y_original, given that the x‑ r accounting for horizontal changes. If no horizontal stretch is present, simply compare y‑values at the same x.
要找到垂直拉伸因子 a，在变换后的图像上选取一个明确的点（不在轴上），找出原函数上相应的点，并在考虑水平变化后 x 坐标匹配的情况下计算 a = y_像点 / y_原点。如果没有水平拉伸，只需比较相同 x 处的 y 值。
For horizontal stretch factor b, compare x‑coordinates for the same y‑ eriodic graphs, using the period change is often easier: b = original period ÷ new period.
对于水平拉伸因子 b，比较同一 y 输出对应的 x 坐标。如果原在 x₀ 出现的特征现在出现在 x₁，那么 b = x₀ / x₁（因为 x₁ = x₀ / b）。在周期图上，利用周期变化通常更容易：b = 原周期 ÷ 新周期。
in(3x) combines vertical stretch by 2 and horizontal compression by 3. Identify each independently by looking at amplitude changes and period changes.
始终检查是否同时应用了垂直与水平拉伸。像 y = 2 sin(3x) 这样的变换图形结合了因子 2 的垂直拉伸和因子 3 的水平压缩。通过分别观察振幅变化和周期变化，独立识别每一个。
10. Stretch Transformations in Standard IB Functions | IB 常见函数中的拉伸变换
tic, cubic, reciprocal, exponential, logarithmic, and trigonometric functions. For y = a·bx + c, the coefficient a acts as a vertical stretch from the basic exponential bx. For y = a log(x - h) + k, a vertically stretches the logarithmic curve.
在 IB 课程中，拉伸常与二次、三次、倒数、指数、对数及三角函数一起出现。对于 y = a·bx + c，系数 a 起着对基本指数函数 bx 的垂直拉伸作用。对于 y = a log(x - h) + k，a 垂直拉伸对数曲线。
e asymptotes shift accordingly: vertical asymptote at x = c/b, horizontal asymptote y = d.
像 y = a / (b x - c) + d 这样的有理函数同时涉及水平和垂直拉伸。可改写为 y = a / [b (x - c/b)] + d，显示出垂直拉伸 a 和水平压缩 b。渐近线相应移动：垂直渐近线 x = c/b，水平渐近线 y = d。
When dealing with trigonometric graphs, remember that vertical
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