2.7.1 Vertical and Horizontal Translations

Understanding how adding or subtracting constants shifts function graphs vertically and horizontally.

定义

Vertical and Horizontal Translations are transformations that shift the graph of a function without changing its shape or orientation. A vertical translation shifts the graph up or down by adding or subtracting a constant to the function value, while a horizontal translation shifts the graph left or right by adding or subtracting a constant to the input variable. Specifically, for a function \(f(x)\): a vertical translation by \(k\) units is represented as \(f(x) + k\) (upward if \(k > 0\), downward if \(k < 0\)), and a horizontal translation by \(h\) units is represented as \(f(x - h)\) (rightward if \(h > 0\), leftward if \(h < 0\)). These translations preserve the domain and range relationships while shifting the location of key features such as intercepts, vertices, and asymptotes.

核心公式

• \(g(x) = f(x) + k\) (vertical translation by \(k\) units)
• \(g(x) = f(x - h)\) (horizontal translation by \(h\) units)
• \(g(x) = f(x - h) + k\) (combined vertical and horizontal translation)
• \(\text{Vertex of } f(x) = a(x - h)^2 + k \text{ is at } (h, k)\)
• \(\text{For } f(x - h) + k: \text{ shift right } h \text{ units and up } k \text{ units}\)

易错点

• ⚠️ Confusing the direction of horizontal translation: students often think \(f(x + h)\) shifts the graph right when it actually shifts left by \(h\) units. The key is remembering that the input value needed to produce the original output is now \(h\) units larger, so the graph moves left.
• ⚠️ Mixing up which constant causes which translation: adding/subtracting outside the function (to the entire function) causes vertical shifts, while adding/subtracting inside the function (to the input) causes horizontal shifts.
• ⚠️ Incorrectly applying translations to asymptotes: students forget that vertical asymptotes shift horizontally with the function, while horizontal asymptotes shift vertically. For example, if \(f(x)\) has a vertical asymptote at \(x = 0\), then \(f(x - 3)\) has a vertical asymptote at \(x = 3\).
• ⚠️ Assuming translations affect the domain and range in unexpected ways: students may incorrectly think that translations change the shape of the domain/range intervals, when in fact they only shift them. For instance, if \(f(x)\) has domain \([0, \infty)\), then \(f(x - 2)\) has domain \([2, \infty)\).