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3.06 Review: 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(x)=2|x|. Plot on the same set of axes and state the domain and range of both $f$f and $f_2(x)$f2(x).

Think: We are vertically dilating $f(x)$f(x) by a factor of $2$2 and reflecting in the x-axis to get $f_2(x)$f2(x). How will this change the key features of the graph?

Do: 

Key Feature $f(x)$f(x) $f_2(x)$f2(x)
Vertex $\left(0,0\right)$(0,0) $\left(0,0\right)$(0,0)
Opening Upwards Downwards
Slope for $x<0$x<0 $-1$1 $2$2
Slope for $x>0$x>0 $1$1 $-2$2
Domain $[-3,3]$[3,3] $[-3,3]$[3,3]
Range $[0,3]$[0,3] $[-6,0]$[6,0]

 

Question 4

How would the graphs of $f(x)=2|x|$f(x)=2|x|,  $f_1(x)=|2x|$f1(x)=|2x| and $f_2(x)=|-2x|$f2(x)=|2x| compare?

Think: We can manipulate these algebraically to compare. 

Do: 

$f_1(x)$f1(x) $=$= $\left|2x\right|$|2x|
  $=$= $\left|2\right|\left|x\right|$|2||x|
  $=$= $2\left|x\right|$2|x|
  $=$= $f(x)$f(x)
$f_2(x)$f2(x) $=$= $\left|-2x\right|$|2x|
  $=$= $\left|-2\right|\left|x\right|$|2||x|
  $=$= $2\left|x\right|$2|x|
  $=$= $f(x)$f(x)

Reflect: These three functions will all actually have the same graph.

 

Question 5

Graph $g(x)=\left|-3x-2\right|$g(x)=|3x2|. State the key features including the domain and range using interval notation.

Think: We can think about this two ways. We can think of this as the graph of $y=-3x-2$y=3x2 with all values below the $x$x-axis reflected to become positive. Or we can think $g(x)$g(x) as a transformation of the form $g(x)=a\left|b\left(x-h\right)\right|+k$g(x)=a|b(xh)|+k. Let's use the second approach.

Do: We need to factor the inside of the absolute value to get it to this form. From there, we can find the key features and graph.

$g(x)$g(x) $=$= $\left|-3x-2\right|$|3x2|
  $=$= $\left|-3\left(x+\frac{2}{3}\right)\right|$|3(x+23)|
  $=$= $\left|-3\right|\left|x+\frac{2}{3}\right|$|3||x+23|
  $=$= $3\left|x+\frac{2}{3}\right|$3|x+23|

So this is the absolute value function by dilated vertically by a stretch factor of $3$3 and translated left $\frac{2}{3}$23 of a unit.

Key Feature $g(x)=3\left|x+\frac{2}{3}\right|$g(x)=3|x+23|
Vertex $\left(\frac{-2}{3},0\right)$(23,0)
Opening Upwards
Slope for $x<\frac{-2}{3}$x<23 $-3$3
Slope for $x>\frac{-2}{3}$x>23 $3$3
Domain $(-\infty,\infty)$(,)
Range $[0,\infty)$[0,)

Reflect: If we had graphed $y=-3x-2$y=3x2 and reflected all negative values in the $x$x-axis we would have had a $y$y-intercept of $y=-2$y=2 become a $y$y-intercept of $y=2$y=2 and a graph of slope $-3$3 which becomes $-3$3 and $3$3 respectively on either side of the vertex.

 

Practice questions

Question 6

Consider the function $y=\left|x+2\right|$y=|x+2|.

  1. Determine the coordinates of the $y$y-intercept.

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

  2. State the coordinate of the vertex.

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

  3. Draw the graph of the function.

    Loading Graph...

Question 7

If the graph of $y=\left|x\right|$y=|x| is stretched vertically by a factor of $4$4 and reflected across the $x$x-axis, what is the equation of the new graph?

Question 8

Graph $y=-\frac{1}{4}\left|x+2\right|$y=14|x+2| $+$+ $4$4.

  1. Loading Graph...

Question 9

Consider the function $y=\left|x-3\right|$y=|x3|.

  1. What is the least possible value that this function can have?

    $0$0

    A

    $3$3

    B

    There is no least value.

    C

    $-3$3

    D
  2. What is the greatest possible value that this function can have?

    $-3$3

    A

    There is no greatest value.

    B

    $0$0

    C

    $3$3

    D
  3. What is the range of the function?

    $y\ge0$y0

    A

    All real $y$y

    B

    $x>3$x>3

    C

    All real $x$x

    D
  4. What is the domain of the function?

    $x<3$x<3

    A

    All real $x$x

    B

    $x\ge0$x0

    C

    $x>0$x>0

    D

 

Outcomes

II.F.IF.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

II.F.IF.7.b

Graph piecewise-defined functions and absolute value functions. Compare and contrast absolute value and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions.

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