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

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

Graph:

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

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

    $x>0$x>0

    $y$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?

Do: 

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(