Single Transformations of Functions

Introduction

One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function $g(x)=f(x)+k$, the function $f(x)$ is shifted vertically k units. For example, $g(x)=x^{2}+4$ is the function $f(x)=x^{2}$ shifted up by 4 units. $g(x)=sin(x)-7.7$ is the function $f(x)=sin(x)$ shifted down by 7.7 units.

Given a function f, a new function $g(x)=f(x-h)$, where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, $g(x)=(x-3)^{2}$ is the graph of $f(x)=x^{2}$ shifted 3 units to the right. $g(x)=sin(x+{\frac {\pi }{2}})$ is the function $f(x)=sin(x)$ shifted ${\frac {\pi }{2}}$ units to the left.

Resources

• Intro to Transformations of Functions, Lumen Learning