Translating Graphs of Functions | 函数图像的平移变换规律

📚 Translating Graphs of Functions | 函数图像的平移变换规律

When a function is transformed by a translation, every point on its graph moves by the same horizontal and vertical displacement. The shape of the curve does not change, but its position on the coordinate plane does. Understanding these rules is essential for sketching graphs quickly and for analysing transformations in algebra and calculus.

当函数图像发生平移变换时，图像上的每一个点都会沿水平和竖直方向移动相同的距离。曲线的形状不会改变，但它在坐标系中的位置会改变。掌握这些规则对于快速作图以及在代数和微积分中分析变换至关重要。

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1. The Basic Principle of Translation | 平移的基本原理

A translation can be described by a vector notation. If a graph is shifted by \( a \) units horizontally and \( b \) units vertically, we can write the translation vector as:

平移可以用向量记号来描述。如果图像在水平方向移动 \( a \) 个单位，在竖直方向移动 \( b \) 个单位，我们可以把平移向量写作：

translation vector = \(\binom{a}{b}\)

Here \( a \) is positive for a shift to the right and negative for a shift to the left; \( b \) is positive for a shift upwards and negative for a shift downwards. This is the foundation of all graph translation questions.

其中 \( a \) 为正表示向右平移，为负表示向左平移；\( b \) 为正表示向上平移，为负表示向下平移。这是解决所有图像平移问题的基础。

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2. Vertical Translation (Adding or Subtracting Outside) | 垂直平移（在函数外加减常数）

If we add a positive constant \( k \) to the whole function, the graph moves upward by \( k \) units.

如果我们将一个正常数 \( k \) 加到整个函数上，图像就向上平移 \( k \) 个单位。

y = f(x) + k → shift upward by k units

If we subtract \( k \), the graph moves downward by \( k \) units.

如果将 \( k \) 减去，图像就向下平移 \( k \) 个单位。

y = f(x) − k → shift downward by k units

For example, the graph of \( y = x^2 + 3 \) is the parabola \( y = x^2 \) shifted 3 units upward. The vertex moves from (0, 0) to (0, 3).

例如，\( y = x^2 + 3 \) 的图像是将抛物线 \( y = x^2 \) 向上平移 3 个单位。顶点从 (0, 0) 移动到 (0, 3)。

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3. Horizontal Translation (Adding or Subtracting Inside) | 水平平移（在函数内加减常数）

If we replace \( x \) by \( x – h \), the graph shifts to the right by \( h \) units. If we replace \( x \) by \( x + h \), the graph shifts to the left by \( h \) units.

如果我们用 \( x – h \) 替换 \( x \)，图像向右平移 \( h \) 个单位。如果我们用 \( x + h \) 替换 \( x \)，图像向左平移 \( h \) 个单位。

y = f(x − h) → shift right by h units

y = f(x + h) → shift left by h units

The direction appears to be reversed because the translation is applied to the input variable \( x \). In the equation \( y = f(x – h) \), we need a larger \( x \) to obtain the same output as before.

方向看起来相反，是因为平移作用在自变量 \( x \) 上。在方程 \( y = f(x – h) \) 中，我们需要更大的 \( x \) 才能得到与原来相同的输出值。

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4. Why the Horizontal Shift Is Reversed | 为什么水平平移方向相反

Consider the point \((0, f(0))\) on the original curve. Under the transformation \( y = f(x – 2) \), this point appears when \( x – 2 = 0 \), so \( x = 2 \). The original output at \( x = 0 \) now occurs at \( x = 2 \). This is why \( f(x – 2) \) moves the graph 2 units to the right.

考虑原曲线上的一点 \((0, f(0))\)。在变换 \( y = f(x – 2) \) 下，这个点出现在 \( x – 2 = 0 \) 时，即 \( x = 2 \)。原来在 \( x = 0 \) 处得到的输出值现在出现在 \( x = 2 \) 处。这就是为什么 \( f(x – 2) \) 使图像向右移动 2 个单位。

Another way to remember this is: to achieve the same y-value, we must compensate inside the bracket. If the bracket contains \( x – 2 \), we need \( x \) to be 2 larger. So the graph moves right.

另一种记忆方法是：为了得到相同的 y 值，我们必须在括号内进行补偿。如果括号中是 \( x – 2 \)，我们需要 \( x \) 比原来大 2，因此图像向右移动。

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5. Combined Translations | 组合平移

When a graph is translated both horizontally and vertically, we can write the transformed function as:

当图像同时进行水平和竖直平移时，我们可以把变换后的函数写成：

y = f(x − h) + k

This means the graph of \( y = f(x) \) is shifted \( h \) units horizontally and \( k \) units vertically. If \( h \) is positive the shift is to the right; if \( h \) is negative, the shift is to the left. If \( k \) is positive the shift is upward; if \( k \) is negative, the shift is downward.

这意味着 \( y = f(x) \) 的图像在水平方向平移 \( h \) 个单位，在竖直方向平移 \( k \) 个单位。若 \( h \) 为正则向右平移；若 \( h \) 为负则向左平移。若 \( k \) 为正则向上平移；若 \( k \) 为负则向下平移。

For example, \( y = (x – 4)^2 + 5 \) is the graph of \( y = x^2 \) shifted 4 units to the right and 5 units upward. Its vertex is \((4, 5)\).

例如，\( y = (x – 4)^2 + 5 \) 是 \( y = x^2 \) 的图像向右平移 4 个单位、向上平移 5 个单位得到的。它的顶点是 \((4, 5)\)。

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6. Translating Specific Points | 平移特殊点

If a point \((x, y)\) lies on the graph of \( y = f(x) \), then after the transformation \( y = f(x – h) + k \), the corresponding point becomes \((x + h, y + k)\).

如果点 \((x, y)\) 在 \( y = f(x) \) 的图像上，那么在变换 \( y = f(x – h) + k \) 之后，对应点变为 \((x + h, y + k)\)。

(x, y) → (x + h, y + k)

This point-by-point rule is useful when you only know a few key points on a graph, such as the vertex of a parabola, the intercepts, or turning points.

这种逐点规则在只知道图像上少数关键点时非常有用，例如抛物线的顶点、截距或驻点。

• Turning points move by the same vector as the whole graph.

  驻点与整个图像一样沿相同向量移动。

• Axis intercepts can be found by substituting \( x = 0 \) for the y-intercept and \( y = 0 \) for the x-intercepts after the transformation.

  截距可以通过在变换后的函数中代入 \( x = 0 \) 求 y 截距、代入 \( y = 0 \) 求 x 截距来得到。

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7. Effect on Domain and Range | 对定义域和值域的影响

A horizontal translation changes the domain of a function. If the original domain is \( a \le x \le b \), then after shifting right by \( h \) units, the new domain is \( a + h \le x \le b + h \).

水平平移会改变函数的定义域。如果原来的定义域是 \( a \le x \le b \)，那么在向右平移 \( h \) 个单位后，新的定义域是 \( a + h \le x \le b + h \)。

A vertical translation changes the range. If the original range is \( c \le y \le d \), then after shifting up by \( k \) units, the new range is \( c + k \le y \le d + k \).

竖直平移会改变函数的值域。如果原来的值域是 \( c \le y \le d \)，那么在向上平移 \( k \) 个单位后，新的值域是 \( c + k \le y \le d + k \)。

If the function has a restricted domain, be sure to translate the endpoints as well as the curve.

如果函数有受限的定义域，请记得端点和曲线一样也要平移。

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8. Effect on Asymptotes and Key Features | 对渐近线和关键特征的影响

Vertical asymptotes move with horizontal translations. If \( y = f(x) \) has a vertical asymptote at \( x = c \), then \( y = f(x – h) \) has the asymptote at \( x = c + h \).

竖直渐近线随水平平移而移动。如果 \( y = f(x) \) 有一条竖直渐近线 \( x = c \)，那么 \( y = f(x – h) \) 的渐近线在 \( x = c + h \)。

Horizontal asymptotes move with vertical translations. If \( y = f(x) \) has a horizontal asymptote at \( y = d \), then \( y = f(x) + k \) has the asymptote at \( y = d + k \).

水平渐近线随竖直平移而移动。如果 \( y = f(x) \) 有一条水平渐近线 \( y = d \)，那么 \( y = f(x) + k \) 的渐近线在 \( y = d + k \)。

For example, \( y = \frac{1}{x} \) has asymptotes \( x = 0 \) and \( y = 0 \). The graph of \( y = \frac{1}{x – 2} + 3 \) has asymptotes \( x = 2 \) and \( y = 3 \).

例如，\( y = \frac{1}{x} \) 有渐近线 \( x = 0 \) 和 \( y = 0 \)。而 \( y = \frac{1}{x – 2} + 3 \) 的渐近线为 \( x = 2 \) 和 \( y = 3 \)。

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9. Worked Example: Parabola | 典例：抛物线

Consider the quadratic function \( f(x) = x^2 \). Find the equation of the graph after translating it 3 units left and 2 units down.

考虑二次函数 \( f(x) = x^2 \)。将它的图像向左平移 3 个单位、向下平移 2 个单位，求变换后的方程。

A shift left by 3 means we replace \( x \) with \( x + 3 \). A shift down by 2 means we subtract 2 from the whole function.

向左平移 3 个单位意味着用 \( x + 3 \) 替换 \( x \)。向下平移 2 个单位意味着整个函数减去 2。

y = (x + 3)^2 − 2

The original vertex is at \((0,0)\). After this translation, the vertex is at \((-3,-2)\).

原来抛物线的顶点在 \((0,0)\)。经过这次平移后，顶点在 \((-3,-2)\)。

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10. Worked Example: Rational Function | 典例：分式函数

Let \( g(x) = \frac{1}{x} \). Find the equation after translating the graph 4 units right and 5 units up.

设 \( g(x) = \frac{1}{x} \)。将图像向右平移 4 个单位、向上平移 5 个单位，求变换后的方程。

Horizontal shift right by 4: replace \( x \) with \( x – 4 \). Vertical shift up by 5: add 5 to the whole function.

向右平移 4 个单位：用 \( x – 4 \) 替换 \( x \)。向上平移 5 个单位：整个函数加 5。

y = \frac{1}{x – 4} + 5

The vertical asymptote changes from \( x = 0 \) to \( x = 4 \). The horizontal asymptote changes from \( y = 0 \) to \( y = 5 \).

竖直渐近线从 \( x = 0 \) 变为 \( x = 4 \)。水平渐近线从 \( y = 0 \) 变为 \( y = 5 \)。

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11. Common Mistakes | 常见错误

Many students incorrectly apply horizontal translations in the same direction as the sign inside the bracket. Remember that \( f(x – h) \) moves right, not left.

许多学生会错误地把括号内符号的方向当作水平平移的方向。请记住 \( f(x – h) \) 是向右平移，而不是向左。

Another common mistake is applying the vertical translation before the horizontal translation, or forgetting to translate every point including asymptotes and domain endpoints.

另一个常见错误是先做竖直平移再做水平平移，或者忘记平移包括渐近线和定义域端点在内的所有点。

• Do not confuse \( y = f(x) + k \) with \( y = f(x + k) \). The first is vertical; the second is horizontal.

  不要混淆 \( y = f(x) + k \) 与 \( y = f(x + k) \)。前者是竖直平移，后者是水平平移。

• When completing the square on a quadratic, use the vertex form to read off the translations correctly.

  对二次函数配方时，利用顶点式来正确读出平移量。

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12. Summary and Exam Tips | 总结与备考提示

In summary, vertical translations are written outside the function, and horizontal translations are written inside the function. Horizontal shifts appear in the opposite direction to the sign inside the bracket.

总之，竖直平移写在函数外面，水平平移写在函数里面。水平平移的方向与括号内符号的方向相反。

y = f(x − h) + k

This single form can represent any translation of a function. In exams, always label at least one key point, such as a vertex, intercept or turning point, to show that the translation has been applied correctly.

这一个式子就可以表示函数的任意平移变换。在考试中，务必标出至少一个关键点，例如顶点、截距或驻点，以说明平移已被正确应用。

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