Virginia SOL Algebra 2 - 2020 Edition
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1.08 Graphs and characteristics of absolute value functions

The characteristics of the parent absolute value function

In our lesson on transformations, we were briefly introduced to the parent absolute value function, $f(x)=\left|x\right|$f(x)=|x|. Below is a bit more detail about this parent function.

Table of values:

$x$x $-2$2 $-1$1 $0$0 $1$1 $2$2
$f(x)=\left|x\right|$f(x)=|x| $2$2 $1$1 $0$0 $1$1 $2$2



Key characteristics:

$y$y-intercept: $\left(0,0\right)$(0,0)

$x$x-intercept: $\left(0,0\right)$(0,0)

Vertex: $\left(0,0\right)$(0,0)

Line of symmetry: $x=0$x=0

Slope for $x<0$x<0 (left of vertex): $-1$1

Slope for $x>0$x>0 (right of vertex): $1$1

Domain: $x$x is any real number, $x\in(-\infty,\infty)$x(,)

Range: $y\ge0$y0, $y\in[0,\infty)$y[0,)


Practice question

Question 1

Consider the function $y=\left|x\right|$y=|x|.

  1. Complete the table.

    $x$x $-2$2 $-1$1 $0$0 $1$1 $2$2
    $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Hence sketch a graph of the function.

    Loading Graph...

  3. State the equation of the axis of symmetry.

  4. State the coordinates of the vertex.

    Vertex $=$=$\left(\editable{},\editable{}\right)$(,)

  5. Write the equation and slope for the two lines that make up the graph of the function.

    Equation Slope
    $x<0$x<0 $y$y$=$=$\editable{}$ $\editable{}$


    $y$y$=$=$\editable{}$ $\editable{}$



Transformations of $f(x)=\left|x\right|$f(x)=|x|

Recall from our lesson on transformations that we can have dilations, reflections and translations done to our parent function.

We will look at transformation of $f(x)=\left|x\right|$f(x)=|x| to give us $g(x)=a\left|b\left(x-h\right)\right|+k$g(x)=a|b(xh)|+k.

Recall the transformations resulting from each coefficient or constant:

$a$a: vertical dilation (stretch if $a>1$a>1 and compression if $00<a<1) and vertical reflection over $x$x-axis if $a<0$a<0

$b$b: horizontal dilation (compression if $b>1$b>1 and stretch if $00<b<1) and horizontal reflection over $y$y-axis if $b$b<0

$h$h: horizontal translation (right if $h>0$h>0 and left if $h<0$h<0)

$k$k: vertical translation (up if $k>0$k>0 and down if $k<0$k<0)


Worked examples

Question 2

Let $f(x)=|x|$f(x)=|x| be defined on the interval $[-3,3]$[3,3]. Construct a new function $f_1(x)=f(x)+1$f1(x)=f(x)+1. That is, $f_1(x)=|x|+1$f1(x)=|x|+1. Plot on the same set of axes and state the domain and range of both $f$f and $f_1(x)$f1(x).

Think: We are translating $f(x)$f(x) up $1$1 unit to get $f_1(x)$f1(x). How will this change the key features of the graph?


Key Feature $f(x)$f(x) $f_1(x)$f1(x)
$x$x-intercept $\left(0,0\right)$(0,0) none
$y$y-intercept $\left(0,0\right)$(0,0) $\left(0,1\right)$(0,1)
Vertex $\left(0,0\right)$(0,0) $\left(0,1\right)$(0,1)
Domain $[-3,3]$[3,3] $[-3,3]$[3,3]
Range $[0,3]$[0,3] $[1,4]$[1,4]


Question 3

Let $f(x)=|x|$f(x)=|x| be defined on the interval $[-3,3]$[3,3]. A new function $f_2$f2 is defined by $f_2(x)=-2|x|$f2(