📚 Investigation 1 - Graphs of Exponential Functions | 探究1：指数函数的图像
Exponential functions are among the most important families of functions in mathematics, underpinning models of growth, decay, and many natural phenomena. In this investigation, we explore how the graph of an exponential function changes with different bases and transformations, using algebraic analysis and graphing technology. Understanding these graphs equips you to solve real‑world problems and tackle IB examination questions with confidence.
指数函数是数学中最重要的函数族之一，是增长、衰减及许多自然现象模型的基础。本次探究中，我们将利用代数分析和绘图技术，探讨指数函数的图像如何随底数和变换而改变。理解这些图像能帮助你解决现实世界的问题，并自信地应对IB考试题目。
1. Defining Exponential Functions | 指数函数的定义
An exponential function is any function of the form y = aˣ function is defined for all real x, while a ≠ 1 avoids the trivial constant function y = 1ˣ = 1.
指数函数是形如 y = aˣ 的任何函数，其中底数 a 是一个正常数且 a ≠ 1。自变量 x 出现在指数位置，这使该函数族区别于幂函数。限制 a > 0 保证了函数对所有实数 x 都有定义，而 a ≠ 1 则避免退化为常值函数 y = 1ˣ = 1。
0, the range depends on the transformations applied. The simplest exponential graph, y = aˣ, has a horizontal asymptote at y = 0 and passes through (0,1) since a⁰ = 1 for any a ≠ 0.
定义域均为全体实数，如果 a > 0，值域则取决于所施加的变换。最简单的指数图像 y = aˣ 有一条水平渐近线 y = 0，且由于 a⁰ = 1 对任何 a ≠ 0 成立，图像总经过点 (0,1)。
2. Graph of y = aˣ for a > 1 | a > 1 时 y = aˣ 的图像
When the base a is greater than 1, the function y = aˣ sents exponential growth. The graph rises from left to right, starting very close to the x‑ tive x. All such graphs share the key point (0,1) and have the horizontal asymptote y = 0.
当底数 a 大于 1 时，函数 y = aˣ 表示指数增长。图像从左到右上升：当 x 为负数时图像非常接近 x 轴，当 x 为正数时则迅速增加。所有此类图像共享关键点 (0,1)，并具有水平渐近线 y = 0。
increases, the graph becomes steeper for x > 0 and flatter for x < 0. For instance, y = 3ˣ grows faster than y = 2ˣ, but both pass through (0,1) and approach zero as x → −∞.
随着 a 增大，图像在 x > 0 时变得更陡峭，在 x < 0 时变得更平坦。例如，y = 3ˣ 比 y = 2ˣ 增长得更快，但两者都通过 (0,1) 并且当 x → −∞ 时趋近于零。
Using a GDC to plot y = 2ˣ, y = 3ˣ and y = 10ˣ on the same axes reveals that they all lie above the x‑axis, are increasing, and intersect the y‑axis at 1. The larger the base, the more dramatically y explodes for positive x.
用图形计算器在同一坐标系中绘制 y = 2ˣ、y = 3ˣ 和 y = 10ˣ，可以看到它们都位于 x 轴上方、单调递增且与 y 轴相交于点 (0,1)。底数越大，y 在正 x 处的爆发式增长就越剧烈。
3. Graphs for 0 < a < 1 | 底数介于 0 和 1 之间的图像
If the base satisfies 0 < a < 1, the function y = aˣ approaching the horizontal asymptote y = 0 as x → +∞, while becoming very large as x → −∞.
如果底数满足 0 < a < 1，函数 y = aˣ 模拟指数衰减。图像严格递减，同样经过点 (0,1)，当 x → +∞ 时趋近水平渐近线 y = 0，而当 x → −∞ 时函数值变得非常大。
Graphs of this type can be obtained by reflecting the graph of y = (1/a)ˣ across the y‑axis. For example, y = (½)ˣ is the reflection of y = 2ˣ over the y‑axis, which is equivalent to y = 2⁻ˣ.
这类图像可以通过将 y = (1/a)ˣ 的图像关于 y 轴反射得到。例如，y = (½)ˣ 是 y = 2ˣ 关于 y 轴的反射，也就等同于 y = 2⁻ˣ。
e smaller the base, the faster the function decays to zero as x increases. Compare y = 0.7ˣ and y = 0.2ˣ arply towards the asymptote. This symmetry property is a powerful tool in graph sketching.
底数越小，随着 x 增大函数衰减到零的速度越快。对比 y = 0.7ˣ 和 y = 0.2ˣ：后者向渐近线下降得更剧烈。这一对称性质是图像绘制中的有力工具。
4. Asymptotes and Intercepts | 渐近线与截距
Every basic exponential graph y = aˣ = 0. There is no vertical asymptote because the function is defined for all real x. The y‑intercept is always (0,1), and there is no x‑intercept because aˣ is never zero.
每一个基本指数图像 y = aˣ（a > 0, a ≠ 1）都有一条水平渐近线 y = 0。由于函数对所有实数 x 都有定义，因此不存在垂直渐近线。y 截距总是 (0,1)，且没有 x 截距，因为 aˣ 永远不等于零。
e horizontal asymptote shifts vertically with any vertical translation. For example, y = 2ˣ tote from y = 0 to y = 3. However, vertical stretches do not affect the asymptote; y = 5·2ˣ still has asymptote y = 0.
当施加变换时，水平渐近线会随着任何垂直平移而垂直移动。例如，y = 2ˣ + 3 将渐近线从 y = 0 移到 y = 3。然而，垂直拉伸不影响渐近线；y = 5·2ˣ 的渐近线仍然为 y = 0。
Reflections can also flip the approach direction. The graph of y = −2ˣ is reflected over the x‑ pproaches y = 0 from below and has the same asymptote y = 0, but now the range is (−∞, 0).
反射也能改变趋近方向。y = −2ˣ 的图像关于 x 轴反射，因此它从下方趋近 y = 0，渐近线仍为 y = 0，但此时值域变成了 (−∞, 0)。
5. Transformations: Vertical Translation | 变换：垂直平移
Adding a constant c to the basic exponential function produces a vertical shift: y = aˣ < 0, they move down. The new asymptote is y = c, and the point (0,1) shifts to (0, 1 + c).
在基本指数函数上加上常数 c 会产生垂直平移：y = aˣ + c。若 c > 0，图像及其渐近线均向上移动 c 个单位；若 c < 0，则向下移动。新的渐近线为 y = c，点 (0,1) 移动到 (0, 1 + c)。
For instance, y = 2ˣ − 4 shifts the entire graph downwards by 4, placing the asymptote at y = −4. The y‑ (0, −3). The shape of the curve remains unchanged; only its vertical position is altered.
例如，y = 2ˣ − 4 将整个图像向下平移 4 个单位，使渐近线位于 y = −4，y 截距变为 (0, −3)。曲线的形状保持不变，仅改变了垂直位置。
the graph is not reflected. Always consider the sign of the vertical translation when determining inequalities.
在草图中标注新的渐近线至关重要，并注意定义域保持为 (−∞, ∞)，而值域对 a > 1 变为 (c, ∞)，对 0 < a < 1 且无反射时也为 (c, ∞)。在确定不等式时，始终要考虑垂直平移的符号。
6. Transformations: Horizontal Translation | 变换：水平平移
Replacing x with (x − h) gives y = a⁽ˣ⁻ʰ Unlike vertical translations, the horizontal asymptote y = 0 remains unchanged, and the y‑intercept changes to (0, a⁻ʰ).
将 x 替换为 (x − h) 得到 y = a⁽ˣ⁻ʰ⁾，使图像水平移动 h 个单位。若 h > 0，图像向右移；若 h < 0，则向左移。与垂直平移不同，水平渐近线 y = 0 保持不变，而 y 截距变为 (0, a⁻ʰ)。
For y = 2⁽ˣ⁻³⁾, the whole graph of y = 2ˣ moves 3 units to the right. The point (0,1) lands at (3,1), and the y‑ercept is (0, 2⁻³) = (0, ⅛). The growth behaviour starts later but is otherwise identical.
对于 y = 2⁽ˣ⁻³⁾，y = 2ˣ 的整个图像向右移动 3 个单位。点 (0,1) 落到 (3,1)，而 y 截距为 (0, 2⁻³) = (0, ⅛)。增长行为开始得较晚，但其他方面完全相同。
Horizontal shifts are often combined with vertical shifts. The order of transformations is usually irrelevant, but for clarity IB students should apply horizontal translation before any stretching or reflecting to avoid sign errors.
水平平移经常与垂直平移结合使用。变换的顺序通常无关紧要，但为了清晰起见，IB 学生应先进行水平平移，再进行拉伸或反射，以避免符号错误。
7. Reflections Across the Axes | 关于坐标轴的反射
A negative sign in front of the function reflects the graph across the x‑axis: y = −aˣ The growth direction reverses; for a > 1, the graph now falls from −∞ to 0 as x increases.
函数前面的负号使图像关于 x 轴反射：y = −aˣ。渐近线仍在 y = 0，但值域翻转为 (−∞, 0)。增长方向反转；对于 a > 1，随着 x 增大，图像从 −∞ 上升到 0。
Placing a negative sign in the exponent reflects the graph across the y‑axis: y = a⁻ˣ. This transformation is equivalent to y = (1/a)ˣ wth function into a decay function (and vice versa). The graph still passes through (0,1).
在指数中放置负号会使图像关于 y 轴反射：y = a⁻ˣ。这一变换等价于 y = (1/a)ˣ，并将增长函数变为衰减函数（或反之）。图像仍然通过点 (0,1)。
Combining both reflections, y = −a⁻ˣ derstanding these symmetries helps quickly sketch variations without plotting many points.
同时进行两种反射，即 y = −a⁻ˣ，关于原点对称，相当于旋转 180°。理解这些对称性有助于快速绘制变体图像，而无需描出许多点。
8. Vertical and Horizontal Stretches | 垂直与水平拉伸
Multiplying the function by a constant A yields y = A·aˣ. When A > 1, the graph is stretched vertically away from the x‑axis; when 0 < A < 1, it is compressed towards the x‑axis. The horizontal asymptote remains y = 0, but the y‑intercept moves from (0,1) to (0, A).
将函数乘以常数 A 得到 y = A·aˣ。当 A > 1 时，图像沿垂直方向远离 x 轴拉伸；当 0 < A < 1 时，则向 x 轴压缩。水平渐近线仍为 y = 0，但 y 截距从 (0,1) 移至 (0, A)。
A horizontal stretch or compression arises from scaling the exponent: y = aᵏˣ horizontally. This transformation can also be reinterpreted by changing the base, since aᵏˣ = (aᵏ)ˣ.
水平拉伸或压缩源于对指数进行缩放：y = aᵏˣ。当 k > 1 时，图像水平压缩；当 0 < k < 1 时，则水平拉伸。这一变换也可以通过改变底数来重新解释，因为 aᵏˣ = (aᵏ)ˣ。
Thus, y = 2³ˣ = 8ˣ se change is a useful strategy in examinations to relate new equations to familiar shapes.
因此，y = 2³ˣ = 8ˣ，图像如同一个更陡峭的指数函数。在考试中，确定底数的变化是一种将新方程与熟悉形状联系起来的有效策略。
9. Combined Transformations | 组合变换
The most general transformed exponential function can be written as y = A·a⁽ᵏ⁽ˣ⁻ʰ on and reflection; h shifts horizontally; and c shifts vertically and moves the asymptote.
最一般的变换后指数函数可写为 y = A·a⁽ᵏ⁽ˣ⁻ʰ⁾⁾ + c。每个参数扮演独特角色：A 控制垂直拉伸与反射；k 控制水平拉伸、压缩与反射；h 负责水平平移；c 则进行垂直平移并移动渐近线。
A systematic approach
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