【指數函數圖像：y = aˣ】Exponential Functions and Graphs: y = aˣ

SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20

y = aˣ（a > 0, a ≠ 1）的圖像必定經過 (0, 1)，並且永遠在 x 軸上方，x 軸是它的漸近線。a > 1 時圖像上升（增長）；0 < a < 1 時圖像下降（衰減）。y = a⁻ˣ 的圖像是它沿 y 軸的反射。The graph of y = aˣ (a > 0, a ≠ 1) always passes through (0, 1) and stays above the x-axis, which is its asymptote. If a > 1 it rises (growth); if 0 < a < 1 it falls (decay). The graph of y = a⁻ˣ is its reflection in the y-axis.

用 AI 逐步解指數函數圖像題目 Solve it with AI練習 49 條 DSE 模擬題 Practise 49 DSE-style questions
拖動底數 a：每條 y = aˣ 都經過 (0, 1) 及 (1, a)。a 小於 1 時下降，大於 1 時上升，a = 3 即例題 1。 Drag the base a: every graph y = aˣ passes through (0, 1) and (1, a). Below 1 the curve falls, above 1 it rises, and a = 3 matches Example 1. 自己試試 → Try it yourself →

指數函數圖像重點 Exponential Graph Facts

必經之點Point on every graph(0, 1) and (1, a)
增長／衰減Growth / decaya > 1 increasing; 0 < a < 1 decreasing
漸近線Asymptotey = aˣ 的漸近線是 y = 0；y = aˣ + c 的是 y = cy = 0 for y = aˣ; y = c for y = aˣ + c
反射Reflectiony = a⁻ˣ = (1/a)ˣ (in the y-axis)
增長模型Growth modelN = N₀ aᵗ
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增長與衰減：y = aˣ 及 y = a⁻ˣ Growth and Decay: y = aˣ and y = a⁻ˣ

a > 1 時，y = aˣ 的圖像上升（增長）。y = a⁻ˣ = (1/a)ˣ 的圖像是它沿 y 軸的鏡像，所以下降（衰減）。兩者都經過 (0, 1)。

For a > 1 the graph of y = aˣ rises (growth). The graph of y = a⁻ˣ = (1/a)ˣ is its mirror image in the y-axis, so it falls (decay). Both pass through (0, 1).

拖動 a：藍色的增長曲線與紅色的衰減曲線永遠沿 y 軸互為鏡像。 Drag a: the blue growth graph and the red decay graph are always mirror images in the y-axis. 自己試試 → Try it yourself →
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垂直平移及漸近線 Vertical Shifts and the Asymptote

加上常數，y = aˣ + c，整個圖像向上移 c 個單位。漸近線由 y = 0 變成 y = c，y 截距變成 1 + c。

Adding a constant, y = aˣ + c, moves the whole graph up by c. The asymptote moves from y = 0 to y = c and the y-intercept becomes 1 + c.

拖動 c：紅色虛線的漸近線隨圖像移動，綠色的 y 截距是 1 + c。c = 1 即例題 2。 Drag c: the red dashed asymptote moves with the graph, and the green y-intercept is 1 + c. c = 1 is Example 2. 自己試試 → Try it yourself →
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指數函數圖像例題詳解 Worked Examples

例題 1 Example 1：由一點求 a Find a from a point

y = aˣ 的圖像經過 (2, 9)，求 a，以及 x = −1 時 y 的值。The graph of y = aˣ passes through (2, 9). Find a and the value of y when x = −1.

  1. a² = 9 而 a > 0，所以 a = 3a² = 9 and a > 0, so a = 3
  2. y = 3⁻¹ = 1/3

答案： Answer: a = 3; y = 1/3

例題 2 Example 2：漸近線及截距 Asymptote and intercept

對於 y = 2ˣ + 1，寫出漸近線及 y 截距。For y = 2ˣ + 1, write down the asymptote and the y-intercept.

  1. y = 2ˣ 的圖像向上移 1，所以漸近線由 y = 0 變為 y = 1The graph of y = 2ˣ moves up by 1, so the asymptote moves from y = 0 to y = 1
  2. At x = 0: y = 2⁰ + 1 = 2

答案： Answer: 漸近線 y = 1；y 截距 2Asymptote y = 1; y-intercept 2

例題 3 Example 3：指數增長 Exponential growth

某培養物每 3 小時加倍，起初有 100 個細胞，求 9 小時後的數目。A culture doubles every 3 hours and starts at 100 cells. Find the number after 9 hours.

  1. N = 100 × 2^(t/3)
  2. t = 9: N = 100 × 2³

答案： Answer: 800

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💡 應試陷阱與秘訣 Exam Traps & Secrets

⚠️ aˣ 永遠不會等於 0 aˣ never reaches 0圖像可以無限接近 x 軸，但永不接觸，所以 aˣ = 0 沒有解。The graph gets as close to the x-axis as you like but never touches it, so aˣ = 0 has no solution.
⚠️ 反射的方向 Which direction is the reflection?y = 2⁻ˣ 是 y = 2ˣ 沿 y 軸的反射；y = −2ˣ 才是沿 x 軸的反射。y = 2⁻ˣ is the reflection of y = 2ˣ in the y-axis, while y = −2ˣ is the reflection in the x-axis.
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✏️ 自我測試：互動練習 Self-Test: Interactive Practice

溫習方法：點擊以下任何題目，即時在坐標網格載入並核對計算步驟。How to study: click any question to load it on the coordinate grid and check your steps.

(畫圖 Sketch)在同一組坐標軸上畫 y = 2^x、y = 2^(−x) 及 y = 2^x − 3。Sketch y = 2^x, y = 2^(−x) and y = 2^x − 3 on one set of axes.
(兩點 Two points)y = k·a^x 的圖像經過 (0, 5) 及 (2, 45)，求 k 及 a。The graph of y = k·a^x passes through (0, 5) and (2, 45). Find k and a.
(衰減 Decay)某汽車每年貶值 15%，現價 $200 000，何時價值減半？A car loses 15% of its value each year and costs $200 000 now. When is it worth half?
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指數函數圖像練習題（DSE 模擬題） Exercises: DSE-Style Practice

SketchMath 題庫內的 DSE 風格練習，選擇題及長題目均附解答。 DSE-style practice from the SketchMath question bank, with solutions for every MC and long question.

指數函數及對數函數 · 中四
47 條選擇題 47 MC · 2 條長題目 2 long questions

常見問題 Frequently Asked Questions

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