跳转至

2.7.3 Vertical and Horizontal Stretches/Compressions

Determining how multiplying functions by constants causes vertical stretching or compression and horizontal scaling.

定义

Vertical and horizontal stretches/compressions describe how multiplying a function by constants transforms its graph. A vertical stretch or compression occurs when the output of a function is multiplied by a constant factor, changing the function's height. A horizontal stretch or compression occurs when the input of a function is multiplied by a constant factor, changing the function's width. Specifically, for a function \(f(x)\):

Vertical Transformations: If \(g(x) = a \cdot f(x)\) where \(a > 0\): - If \(a > 1\), the graph is vertically stretched by a factor of \(a\) (becomes taller) - If \(0 < a < 1\), the graph is vertically compressed by a factor of \(a\) (becomes shorter) - If \(a < 0\), the graph is also reflected across the x-axis

Horizontal Transformations: If \(g(x) = f(bx)\) where \(b > 0\): - If \(b > 1\), the graph is horizontally compressed by a factor of \(\frac{1}{b}\) (becomes narrower) - If \(0 < b < 1\), the graph is horizontally stretched by a factor of \(\frac{1}{b}\) (becomes wider) - If \(b < 0\), the graph is also reflected across the y-axis \nThese transformations are fundamental to understanding function behavior and are essential for analyzing and sketching complex functions.

核心公式

  • \(g(x) = a \cdot f(x)\) (vertical stretch/compression by factor \(a\))
  • \(g(x) = f(bx)\) (horizontal compression/stretch by factor \(\frac{1}{b}\))
  • \(g(x) = a \cdot f(bx)\) (combined vertical and horizontal transformation)
  • \(\text{Vertical stretch factor} = |a| \text{ where } a > 1 \text{ stretches, } 0 < a < 1 \text{ compresses}\)
  • \(\text{Horizontal compression factor} = \frac{1}{b} \text{ where } b > 1 \text{ compresses, } 0 < b < 1 \text{ stretches}\)

易错点

  • ⚠️ Confusing the direction of horizontal transformations: students often think that multiplying the input by a value greater than 1 causes a stretch, when it actually causes a compression. The factor is inverted: \(f(2x)\) compresses horizontally by \(\frac{1}{2}\), not stretches by 2.
  • ⚠️ Forgetting that horizontal transformations use the reciprocal factor: when given \(g(x) = f(bx)\), the horizontal scaling factor is \(\frac{1}{b}\), not \(b\). This is counterintuitive compared to vertical transformations.
  • ⚠️ Misidentifying which transformation is applied when given a combined function like \(g(x) = 2f(3x)\): students may incorrectly state that the vertical stretch is 3 and horizontal compression is 2, when actually the vertical stretch is 2 and horizontal compression is \(\frac{1}{3}\).
  • ⚠️ Neglecting the effect of negative coefficients: students often forget that negative values of \(a\) or \(b\) also include a reflection (across the x-axis for \(a < 0\), across the y-axis for \(b < 0\)) in addition to the stretch/compression.