Translating functions

Operations for functions: New functions from old

Translating functions

We will consider various ways of manipulating a graph in the plane and study the consequences in case the graph is the graph of a function. We will begin by shifting the graph up or down.

Vertical translation

Let $a$ be a real number. The vertical translation by $a$ sends the point $\rv{x,y}$ of the plane to $\rv{x,y+a}$ .

Let $f$ be a real function. Under the vertical translation by $a$ , the graph of $f$ is turned into the graph of the function of $x$ with rule $f(x)+a$ .

The graph of the function $f$ moves upwards if $a\gt0$ and downwards if $a\gt0$ .

The location of the zeros of the function $f$ will change but the location of the minima and/or maxima will not change in terms of $x$ but only in terms of $y$ where $a$ has to be added to the original value.

By way of example we have drawn two vertical translations of the graph of $f(x)=x^2-1$ :

The following facts can be observed from these graphs:

The graph of $f$ is a parabola with zeros $x=-1$ and $x=1$ and a minimum at $f(0)=-1$ .
The graph of $h$ is a parabola with zeros $x=\sqrt{3}$ and $x=-\sqrt{3}$ and a minimum at $f(0)=-3$ .
The graph of $g$ is a parabola with no zeros and a minimum at $f(0)=1$ .

Next we deal with shifts of the graph to the left or right.

Horizontal translation

Let $a$ a real number. The horizontal translation by $a$ sends the point $\rv{x,y}$ of the plane to $\rv{x+a,y}$ .

Let $f$ be a real function. Under the horizontal translation by $a$ , the graph of $f$ is turned into the graph of the function of $x$ with rule $f(x-a)$ .

The function $f$ moves to the left if $a\gt0$ and to the right if $a\lt0$ . The location of the zeros of the function $f$ will move in the same direction (by adding $a$ to the value of $x$ ) but the location of the minima and/or maxima will change in terms of $x$ where $a$ has to be added to the original value but the $y$ will remain the same. By way of example we have drawn two vertical translations of the graph of $f(x)=x^2-1$ :

The following facts can be observed from these graphs:

The graph of $f$ is a parabola with zeros $x=-1$ and $x=1$ and a minimum at $f(0)=-1$ .
The graph of $h$ is a parabola with zeros $x=-2$ and $x=0$ and a minimum at $f(-1)=-1$ .
The graph of $g$ is a parabola with zeros $x=0$ and $x=2$ and a minimum at $f(1)=-1$ .

Let $f(x)= {{x-6}\over{x+6}}$ . Below the graph of $f$ is drawn as well as its vertical translation and horizontal translation by $2$ .

Which graph corresponds to which translation?

The graph of $f$ is blue.
The vertical translation of the graph belongs to the function $2+ f(x) = 2 + \left({{x-6}\over{x+6}}\right)$ $={{3\cdot x+6}\over{x+6}}$ is red.
The horizontal translation of the graph belongs to the function $f\left({x- 2}\right) = {{x-8}\over{x+4}}$ $={{x-8}\over{x+4}}$ is green.

New example