两个（或多个）函数的复合仍然是函数

(Video) Translating Absolute Value Functions

by Keith Mann

There are two ways we will learn to translate a function: Horizontal and Vertical. Keith Mann explains both. Skip to $3:20$ for vertical translations. Skip to $6:00$ for horizontal translations.

$\large{\text{Summary}}$

This video explains both vertical $(f(x) = |x| \rightarrow g(x) = |x|-5)$ and horizontal translations $(f(x) = |x| \rightarrow h(x) = |x+1|)$ of absolute value functions.

For a challenge, let's compose these translations so that our new function is shifted $5$ units down and $1$ unit left!

First, let's translate the original absolute value function, $f(x)=|x|$, five units down. This means we move every point on the original absolute function down five units.

Here are the graphs of the functions:

The red function is $f(x)=|x|$ and the blue function is $g(x)=|x|-5.$ Notice that $g(x)$ looks similar to our original function, except it is shifted vertically downward by $5$ units.

We can represent vertical shifts downwards by subtracting a positive number from our function outside of the absolute value signs. We can represent vertical shifts upwards by adding a positive number to our function outside of the absolute value signs.

Now, let's translate out new absolute value function, $g(x)=|x|-5$, one unit to the left. After watching the video, we see that this clearly means we move every point on $g(x)$ to the left by one unit.

The red function, $f(x)=|x|,$ and the blue function, $g(x)=|x|-5,$ are the same as before. We obtained the green function, $h(x)=|x+1|-5$, by horizontally translating $g(x)$ by one unit to the left. Notice that $h(x)$ looks similar to our original functions, except its vertex is at $(-1,-5)$.

When we are working with absolute value functions, we can represent horizontal shifts to the left by adding a positive number to our function inside of the absolute value signs. We can represent horizontal shifts to the right by subtracting a positive number=from our function inside of the absolute value signs.

Function Transformation

A function transformation is to move the graph by shifting the axis of the graph, stretch, or compress the graph.

It take the basic function of $f(x)$ and transforms it, moving the graph around.

There are $6$ common types of function transformations: $$\begin{array}{|r|c|c|} \hline \text{Shift Up or Down:}& f(x)+c \\ \hline \text{Shift Left or Right:}& f(x+c) \\ \hline \text{Reflect about the x-axis:}& -f(x) \\ \hline \text{Reflect about the y-axis:}& f(-x) \\ \hline \text{Stretch or Compress Vertically:}& cf(x) \\ \hline \text{Stretch or Compress Horizontally:}& f(cx)\\ \hline \end{array}$$

Let's take a look at some quadratic functions.

Shift Up or Down

Given the graph of $f(x)$, $f(x)+c$ will be the graph of $f(x)$ shifted up by $c$ if the value of $c$ is positive and shifted down by $c$ if the value of $c$ is negative. $\\$ Let’s start with the graph of $f(x)=x^{2}$. $\\$

The graph $g(x)= f(x)+3 = x^{2}+3$ will be the original graph of $f(x)$ shifted up $3$ units. $\\$

The graph $g(x)= f(x)-3 = x^{2}-3$ will be the original graph of $f(x)$ shifted down $3$ units. $\\$

Shift Left or Right

Given the graph of $f(x)$, $f(x+c)$ will be the graph of $f(x)$ shifted left by $c$ if the value of $c$ is positive and shifted right by $c$ if the value of $c$ is negative. $\\$ Let’s start with the graph of $f(x)=x^{2}$ again. $\\$

The graph $g(x)= f(x+3) = (x+3)^{2}$ will be the original graph of $f(x)$ shifted left by $3$ units. $\\$

The graph $g(x)= f(x-3) = (x-3)^{2}$ will be the original graph of $f(x)$ shifted left by $3$ units. $\\$

Reflect about the x-axis

Given the graph of $f(x)$, $-f(x)$ will be the graph of $f(x)$ reflected about the x-axis. $\\$ Let’s start with the graph of $f(x)=x^{2}$. $\\$

The graph $g(x)=-f(x)=-x^{2}$ will be the original graph of $f(x)$ reflected about the x-axis. $\\$

Reflect about the y-axis

Given the graph of $f(x)$, $f(-x)$ will be the graph of $f(x)$ reflected about the y-axis. $\\$ Let’s start with the graph of $f(x)=\sqrt{x}$. $\\$

The graph $g(x)=f(-x)=\sqrt{-x}$ will be the original graph of $f(x)$ reflected about the y-axis. $\\$

Stretch or Compress Vertically

Given the graph of $f(x)$, $cf(x)$ will be the graph of $f(x)$ stretched vertically if the value of $c$ is $c>1$. $cf(x)$ will be the graph of $f(x)$ compressed vertically if the value of $c$ is $0 <c<1$. $\\$ Let’s start with the graph of $f(x)=x^{2}$. $\\$

The graph $g(x)=2f(x)=2x^{2}$ will be the original graph of $f(x)$ stretched vertically by $2$ units. $\\$

The graph $g(x)=\dfrac{1}{2}f(x)=\dfrac{1}{2}x^{2}$ will be the original graph of $f(x)$ compressed vertically by $2$ units. $\\$

Stretch or Compress Horizontally

Given the graph of $f(x)$, $f(cx)$ will be the graph of $f(x)$ stretched horizontally if the value of $c$ is $0 <c<1$. $f(cx)$ will be the graph of $f(x)$ compressed horizontally if the value of $c$ is $c>1$.

Let’s start with the graph of $f(x)=x^{2}$.

Horizontal stretches:

The graph $g(x)=f(\dfrac{1}{2}x)=(\dfrac{1}{2}x)^{2}$ will be the original graph of $f(x)$ stretched horizontally by $2$ units. $\\$

The graph $g(x)=f(2x)=(2x)^{2}$ will be the original graph of $f(x)$ compressed horizontally by $2$ units. $\\$ The red graph is the graph of $f(x)=x^2$, and the blue graph is the graph of $g(x)=(2x)^{2}$.

General Translations

Here are the types of translations:

• Horizontally left
• Horizontally right
• Vertically down
• Vertically up

The general form of the absolute value equation is $$y=|x|$$

A left translation would be shown by $$y=|x+a| \text{ with } a>0$$

A right translation would be shown by $$y=|x+a| \text{ with } a<0$$

A downward translation would be shown by $$y=|x|+b \text{ with } b<0$$

An upward translation would be shown by $$y=|x|+b \text{ with } b>0$$

Let's look at each translation of $2$ units compared to the original graph of $y=|x|$ drawn in black.

Left translation $2$ units: $$y=|x+2|$$

Right translation $2$ units: $$y=|x-2|$$

Downward translation $2$ units: $$y=|x|-2$$

Upward translation $2$ units: $$y=|x|+2$$

Function Transformation

A function transformation changes the way a function looks on a graph.

There are three main types of transformations, which we will dive into deeper in further lessons.

• Translation: shifts the function up, down, left, or right
• Reflection: flips the function over a line, including the $x$-axis or $y$-axis
• Dilation: stretches a function to make it taller or wider or compresses a function to make it shorter or narrower, which you can do vertically or horizontally

Looking Further

Understanding this concept is important when we tackle more complicated math, like absolute value functions and inverse functions. In many of these more complicated functions, we work from the parent function.