【函數圖像變換：平移、反射及伸縮】Graph Transformations: Translation, Reflection and Stretching of y = f(x)
SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
y = f(x) + k 向上移 k；y = f(x − h) 向右移 h；y = −f(x) 沿 x 軸反射；y = f(−x) 沿 y 軸反射；y = a·f(x) 垂直伸展 a 倍；y = f(bx) 水平壓縮為 1/b。y = f(x) + k shifts up by k; y = f(x − h) shifts right by h; y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis; y = a·f(x) stretches vertically by a; y = f(bx) squeezes horizontally by 1/b.
拖動 h：紅色圖像 y = (x − h)² + 3 是灰色圖像 y = x² 向右移 h 個單位、向上移 3 個單位。h = 2 即例題 1 的頂點 (2, 3)。 Drag h: the red graph y = (x − h)² + 3 is the grey graph y = x² moved h units to the right and 3 units up. h = 2 gives the vertex (2, 3) of Example 1. 自己試試 → Try it yourself →
圖像變換規則 Transformation Rules
上下平移Translation up / downy = f(x) + k
左右平移Translation right / lefty = f(x − h) (right if h > 0)
平移：圖像上下左右移動 Translation: Moving the Graph Up, Down, Left or Right
y = f(x) + k 把圖像向上移 k 個單位（k 為負則向下）。y = f(x − h) 把圖像向右移 h 個單位。每一點都移動相同的距離，形狀不變。
y = f(x) + k moves the graph up by k (down if k is negative). y = f(x − h) moves it right by h. Every point moves the same way, so the shape does not change.
拖動 k：灰色的 y = x² 變成 y = x² + k 上下移動，頂點由 (0, 0) 移到 (0, k)。 Drag k: the grey graph y = x² moves up and down as y = x² + k, and its vertex goes from (0, 0) to (0, k). 自己試試 → Try it yourself →
反射：沿 x 軸或 y 軸翻轉 Reflection: Flipping in the x-axis or the y-axis
y = −f(x) 把圖像沿 x 軸反射（每個 y 變號）。y = f(−x) 把圖像沿 y 軸反射（每個 x 變號）。
y = −f(x) reflects the graph in the x-axis (every y changes sign). y = f(−x) reflects it in the y-axis (every x changes sign).
把 u 由 0 拖到 1：灰色的 y = 2ˣ 沿 x 軸（紅色）及沿 y 軸（綠色）翻轉。 Drag u from 0 to 1: the grey graph y = 2ˣ flips in the x-axis (red) and in the y-axis (green). 自己試試 → Try it yourself →
y = a·f(x) 把圖像垂直伸展 a 倍。y = f(bx) 把圖像水平壓縮為 1/b（與很多同學的直覺相反）。
y = a·f(x) stretches the graph vertically by the factor a. y = f(bx) squeezes it horizontally by the factor 1/b (the opposite of what many students expect).
拖動 s：紅色的 y = s sin x 變得更高，綠色的 y = sin(sx) 則變得更窄。 Drag s: the red graph y = s sin x gets taller, while the green graph y = sin(sx) gets narrower. 自己試試 → Try it yourself →
⚠️ 水平平移的符號是相反的 The sign of the horizontal shift is backwardsf(x − 3) 是向右移 3 個單位，f(x + 3) 是向左移。想想哪個 x 會得出原來的值。f(x − 3) moves the graph 3 units to the right, and f(x + 3) to the left. Think of which x gives the old value.
⚠️ 水平變換是作用在 x 上 Horizontal changes act on x, not on the whole functionf(2x) 把每個點的 x 坐標減半（壓縮為 1/2），與很多同學的直覺相反。f(2x) halves every x-coordinate (a squeeze by 1/2), the opposite of what many students expect.
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.
(組合變換 Combined)y = f(x) 的最低點是 (1, −2)，寫出 y = 3f(x − 2) + 1 的最低點。The graph of y = f(x) has a minimum point at (1, −2). Write the minimum point of y = 3f(x − 2) + 1.
(描述變換 Describe)描述 y = x² 變成 y = x² − 6x + 11 的變換。Describe the transformation from y = x² to y = x² − 6x + 11.
(畫圖 Sketch)以 f(x) = 2^x，畫 y = f(x) 及 y = f(−x) + 2。Sketch y = f(x) and y = f(−x) + 2 for f(x) = 2^x.