1.7 | Transformations of Functions


Let \(c\) be a positive real number. Vertical and horizontal shifts in the graph of \(y = f(x)\) are represented as follows:

  1. Vertical shift \(c\) units upward:
    $$
    h(x) = f(x) + c
    $$
  2. Vertical shift \(c\) units downward:
    $$
    h(x) = f(x) – c
    $$
  3. Horizontal shift \(c\) units to the right:
    $$
    h(x) = f(x-c)
    $$
  4. Horizontal shift \(c\) units to the left:
    $$
    h(x) = f(x+c)
    $$
Reflections in the coordinate axes of the graph of \(y = f(x)\) are represented as follows.

  1. Reflection in the \(x\)-axis:
    $$
    h(x) = -f(x)
    $$
  2. Reflection in the \(y\)-axis:
    $$
    h(x) = f(-x)
    $$
A vertical stretch of the graph \(y = f(x)\) is represented by \(g(x) = c f(x),\) where \(c>1\). A vertical shrink is represented by \(g(x) = c f(x),\) where \( 0 < c < 1\). A horizontal stretch is represented by \(g(x) = f(c x),\) where \(0 < c < 1\). A horizontal shrink is represented by \(g(x)= f(c x),\) where \(c > 1.\)

E 1.7 Exercises