# Horizontal Shift – Definition, Process and Examples

The horizontal shift highlights how the input value of the function affects its graph. When dealing with horizontal shifts, the focus is solely on how the graph and function behave along the $x$-axis. Understanding how horizontal shifts work is important, especially when graphing complex functions.

The horizontal shift occurs when a graph is shifted along the $\boldsymbol{x}$-axis by $\boldsymbol{h}$ units — either to the left or to the right.

Along with other transformations, it is important to know how to identify and apply horizontals on different functions — including trigonometric functions. This article covers all key concepts needed to master this topic!

## What Is a Horizontal Shift?

A horizontal shift is a translation that shifts the function’s graph along the $x$-axis. It describes how it is shifted from one function to the right or to the left to find the position of the new function’s graph. In a horizontal shift, the function $f(x)$ is shifted $h$ units horizontally and results to translating the function to $f(x \pm h)$.

Take a look at the graphs of the three functions: $f(x) = x^2$, $g(x) = (x + 3)^2$, and $h(x) = (x – 3)^2$. With $f(x)$ as the parent function or the basic function of quadratic functions, the two remaining functions are the result of horizontally shifting $f(x)$.

• When $f(x) =x^2$ is shifted $3$ units to the left, this results to its input value being shifted $+3$ units along the $x$-axis. Hence, the translated function is equal to $g(x) = (x- 3)^2$.
• Similarly, when the parent function is shifted $3$ units to the right, the input value will shift $-3$ units horizontally. This results to the translated function $h(x) = (x -3)^2$.

This behavior is true for all horizontal shifts, so it’s best to establish a general rule on what to expect when the function $f(x)$ is shifted $h$ units to the right or $h$ units to the left.

### Rules for the Horizontal Shift

Suppose that $h$ is greater than zero and when $f(x)$ is shifted $h$ units along the $x$-axis, it results in the following functions:

1.      $\boldsymbol{y = f(x – h)}$ : a horizontal shift of $h$ units to the right.

2.     $\boldsymbol{y = f(x + h)}$ : a horizontal shift of $h$ units to the left.

When horizontally shifting a function or its graph, the size and shape of the function remain the same.

To better understand how the coordinates of the function are affected after a horizontal shift, construct a table of values for $f(x) = x^2$, $g(x) = (x + 1)^2$, and $h(x) = (x – 1)^2$.

 \begin{aligned} \boldsymbol{x} \end{aligned} \begin{aligned}-2\end{aligned} \begin{aligned}-1\end{aligned} \begin{aligned}0\end{aligned} \begin{aligned}1\end{aligned} \begin{aligned}2\end{aligned} \begin{aligned} \boldsymbol{y = x^2} \end{aligned} \begin{aligned}4\end{aligned} \begin{aligned}1\end{aligned} \begin{aligned}0\end{aligned} \begin{aligned}1\end{aligned} \begin{aligned}4\end{aligned} \begin{aligned} \boldsymbol{y=(x-1)^2} \end{aligned} \begin{aligned}9\end{aligned} \begin{aligned}4\end{aligned} \begin{aligned}1\end{aligned} \begin{aligned}0\end{aligned} \begin{aligned}1\end{aligned} \begin{aligned} \boldsymbol{y=(x +1)^2} \end{aligned} \begin{aligned}1\end{aligned} \begin{aligned}0\end{aligned} \begin{aligned}1\end{aligned} \begin{aligned}4\end{aligned} \begin{aligned}9\end{aligned}