Transformations of Absolute Value Function: Insights and Applications

Transformations of $y = ∣ x ∣$
Vertical Translations	$Translation up k units, k > 0 y = ∣ x ∣ + k$
$Translation down k units, k < 0 y = ∣ x ∣ + k$
Horizontal Translations	$Translation to the right h units, h > 0 y = ∣ x - h ∣$
$Translation to the left h units, h < 0 y = ∣ x - h ∣$

Transformations of

y = ∣ x ∣

Vertical Translations

Translation up k units, k > 0 y = ∣ x ∣ + k

Translation down k units, k < 0 y = ∣ x ∣ + k

Horizontal Translations

Translation to the right h units, h > 0 y = ∣ x - h ∣

Translation to the left h units, h < 0 y = ∣ x - h ∣

Transformations of $y = f (x)$
Horizontal Translations	$Translation to the right by h units, h > 0 y = f (x - h)$
$Translation to the left by h units, h < 0 y = f (x - h)$
Vertical Translations	$Translation upwards by k units, k > 0 y = f (x) + k$
$Translation downwards by k units, k < 0 y = f (x) + k$

Transformations of

y = f (x)

Horizontal Translations

Translation to the right by h units, h > 0 y = f (x - h)

Translation to the left by h units, h < 0 y = f (x - h)

Vertical Translations

Translation upwards by k units, k > 0 y = f (x) + k

Translation downwards by k units, k < 0 y = f (x) + k

Transformations of $f (x)$
Vertical Stretch or Shrink	Vertical stretch, $a > 1$ $y = a f (x)$
Vertical shrink, $0 < a < 1$ $y = a f (x)$

Transformations of

f (x)

Vertical Stretch or Shrink

Vertical stretch,

a > 1

y = a f (x)

Vertical shrink,

0 < a < 1

y = a f (x)

Transformations of $f (x)$
Horizontal Stretch or Shrink	Horizontal stretch, $0 < b < 1$ $y = f (b x)$
Horizontal shrink, $b > 1$ $y = f (b x)$

Transformations of

f (x)

Horizontal Stretch or Shrink

Horizontal stretch,

0 < b < 1

y = f (b x)

Horizontal shrink,

b > 1

y = f (b x)

Transformations of $y = ∣ x ∣$
Vertical Stretch or Shrink	$Vertical stretch, a > 1 y = a ∣ x ∣$
$Vertical shrink, 0 < a < 1 y = a ∣ x ∣$
Horizontal Stretch or Shrink	$Horizontal stretch, 0 < b < 1 y = ∣ b x ∣$
$Horizontal shrink, b > 1 y = ∣ b x ∣$

Transformations of

y = ∣ x ∣

Vertical Stretch or Shrink

Vertical stretch, a > 1 y = a ∣ x ∣

Vertical shrink, 0 < a < 1 y = a ∣ x ∣

Horizontal Stretch or Shrink

Horizontal stretch, 0 < b < 1 y = ∣ b x ∣

Horizontal shrink, b > 1 y = ∣ b x ∣

Transformations of $f (x)$
Reflections	In the $x -$ axis $y = - f (x)$
In the $y -$ axis $y = f (- x)$

Transformations of

f (x)

Reflections

In the

x -

axis

y = - f (x)

In the

y -

axis

y = f (- x)

Transformations of $y = ∣ x ∣$
Reflections	$In the x - axis y = - ∣ x ∣$
$In the y - axis y = ∣ - x ∣$

Transformations of

y = ∣ x ∣

Reflections

In the x - axis y = - ∣ x ∣

In the y - axis y = ∣ - x ∣

$x$	$f (x) = ∣ 2 x - 5 ∣$	$f (x)$
$- 1$	$f (x) = ∣ 2 (- 1) - 5 ∣$	$7$
$0$	$f (x) = ∣ 2 (0) - 5 ∣$	$5$
$1$	$f (x) = ∣ 2 (1) - 5 ∣$	$3$
$2.5$	$f (x) = ∣ 2 (2.5) - 5 ∣$	$0$
$4$	$f (x) = ∣ 2 (4) - 5 ∣$	$3$
$5$	$f (x) = ∣ 2 (5) - 5 ∣$	$5$
$6$	$f (x) = ∣ 2 (6) - 5 ∣$	$7$

x

f (x) = ∣ 2 x - 5 ∣

f (x)

- 1

f (x) = ∣ 2 (- 1) - 5 ∣

7

0

f (x) = ∣ 2 (0) - 5 ∣

5

1

f (x) = ∣ 2 (1) - 5 ∣

3

2.5

f (x) = ∣ 2 (2.5) - 5 ∣

0

4

f (x) = ∣ 2 (4) - 5 ∣

3

5

f (x) = ∣ 2 (5) - 5 ∣

5

6

f (x) = ∣ 2 (6) - 5 ∣

7

$x$	$f (x) = \frac{1}{2} ∣ x - 4 ∣ - 2$	$f (x)$
$- 2$	$f (x) = \frac{1}{2} ∣ - 2 - 4 ∣ - 2$	$1$
$0$	$f (x) = \frac{1}{2} ∣ 0 - 4 ∣ - 2$	$0$
$2$	$f (x) = \frac{1}{2} ∣ 2 - 4 ∣ - 2$	$- 1$
$4$	$f (x) = \frac{1}{2} ∣ 4 - 4 ∣ - 2$	$- 2$
$6$	$f (x) = \frac{1}{2} ∣ 6 - 4 ∣ - 2$	$- 1$
$8$	$f (x) = \frac{1}{2} ∣ 8 - 4 ∣ - 2$	$0$
$10$	$f (x) = \frac{1}{2} ∣ 10 - 4 ∣ - 2$	$1$

x

f (x) = \frac{1}{2} ∣ x - 4 ∣ - 2

f (x)

- 2

f (x) = \frac{1}{2} ∣ - 2 - 4 ∣ - 2

1

0

f (x) = \frac{1}{2} ∣ 0 - 4 ∣ - 2

0

2

f (x) = \frac{1}{2} ∣ 2 - 4 ∣ - 2

- 1

4

f (x) = \frac{1}{2} ∣ 4 - 4 ∣ - 2

- 2

6

f (x) = \frac{1}{2} ∣ 6 - 4 ∣ - 2

- 1

8

f (x) = \frac{1}{2} ∣ 8 - 4 ∣ - 2

0

10

f (x) = \frac{1}{2} ∣ 10 - 4 ∣ - 2

1

The parent function of an absolute value function is $f (x) = ∣ x ∣$ and its graph is as follows.

Determine the graph of each of the following functions by translating the absolute value parent function.

Let $g (x) = ∣ x + 2 ∣ .$

Which of these is the graph of

g (x) ?

Let $h (x) = ∣ x ∣ - 2 .$

Which of these is the graph of

h (x) ?

Let's recall the given function. g(x)=|x+ 2| We can see that a number is added to the input of f(x) to get g(x). This means that g(x) is the horizontal translation of f(x) to the left 2 units.

Therefore, the graph of g(x) is given in option D.

When we look at the function h(x), we can see that a number is subtracted from the output of f(x). h(x)=|x|- 2 With this operation, we are translating the graph of f(x) down 2 units.

From here, we can conclude that the graph of h(x) is given in option A.

The table below summarizes the different types of translations that can be performed for shifting an absolute value function.

Transformations of y=\|x\|
Vertical Translations	Translation up k units, k>0 y=\|x\|+ k
Vertical Translations	Translation down k units, k<0 y=\|x\|+ k
Horizontal Translations	Translation to the right h units, h>0 y=\|x- h\|
Horizontal Translations	Translation to the left h units, h<0 y=\|x- h\|

We can use this table to identify the translations applied to the absolute value functions.

Write a function that represents the horizontal shrink of the graph of the absolute value parent function

f (x) = ∣ x ∣

by a factor of

3 .

The graph of the absolute value parent function $f (x) = ∣ x ∣$ is given along with the graph of $h (x) = a ∣ x ∣,$ where $a$ is a real number other than $0 .$

Vertical Shrink of Absolute Value Functions

Find the value of

a

by comparing

f (x)

and

h (x) .

Let's begin by recalling the general form of a horizontal stretch or shrink of an absolute value function.

Transformations of y=\|x\|
Horizontal Stretch or Shrink	Horizontal stretch, 0< b<1 y=\| bx\|
Horizontal Stretch or Shrink	Horizontal shrink, b>1 y=\| bx\|

Since we want a horizontal shrink by a factor of 3, we can write the required function by substituting 3 for b in the given form. g(x)=| bx| ⇒ g(x)=| 3x|

Note that we can get the graph of h(x) by multiplying the output of f(x) by a. Therefore, the graph of h(x) is the graph of f(x) vertically stretched or shrunk. Let's choose a point on each graph and compare them vertically.

We can see that the y-coordinate of the point on h(x) is half the y-coordinate of the point on f(x). This means that we can get the y-coordinate of the point on h(x) by multiplying the y-coordinate of the point on f(x) by 12. Let's recall the general form of a vertical stretch or shrink of an absolute value function.

Transformations of y=\|x\|
Vertical Stretch or Shrink	Vertical stretch, a>1 y= a\|x\|
Vertical Stretch or Shrink	Vertical shrink, 0< a< 1 y= a\|x\|

This table tells us that the graph of h(x) is the graph of f(x) vertically shrunk by a factor of 12. Therefore, the value of a is 12.

Write a function

g (x)

whose graph is the reflection of the graph of

f (x) = 3 ∣ x - 2 ∣

in the

y -

axis.

Let's begin by recalling the general form of a reflection of an absolute value function.

Transformations of y=\|x\|
Reflections	In the x-axis y=- \|x\|
Reflections	In the y-axis y=\|- x\|

This table tells us that if we want to reflect an absolute value function in the y-axis, we should multiply its input by -1. ccc Original Function & & Reflection iny-axis [0.5em] f(x)=3|x-2| & & g(x)=3|- x-2| We can simplify the function g(x) by using the properties of absolute value.

The function whose graph is the reflection of the graph of f(x)=3|x-2| in the y-axis is g(x)=3|x+2|.

Consider the graph of the absolute parent function $f (x) = ∣ x ∣ .$

Let the graph of $h (x)$ be the graph of $f (x)$ translated $3$ units to the right, vertically stretched by a factor of $2,$ and reflected on the $x -$ axis.

Which of these graphs is the graph of

h (x) ?

To determine the graph of h(x), we will apply the transformations to the graph of f(x) one at a time. Let's begin by translating the graph of f(x) to the right 3 units. To do so, we add 3 to the x-coordinate of the points on the graph of f(x).

Next, we will stretch the resulting graph vertically by a factor of 2 by multiplying the y-coordinate of the points on the graph by 2.

Finally, we reflect the graph in the x-axis by multiplying the y-coordinate of the points on the graph by -1.

Now that we have the graph of h(x), we can see that it matches the graph in option B.

Transformations of y=\|x\|
Vertical Translations	Translation up k units, k>0 y=\|x\|+ k
Vertical Translations	Translation down k units, k<0 y=\|x\|+ k
Horizontal Translations	Translation to the right h units, h>0 y=\|x- h\|
Horizontal Translations	Translation to the left h units, h<0 y=\|x- h\|

Transformations of y=\|x\|
Horizontal Stretch or Shrink	Horizontal stretch, 0< b<1 y=\| bx\|
Horizontal Stretch or Shrink	Horizontal shrink, b>1 y=\| bx\|

Transformations of y=\|x\|
Vertical Stretch or Shrink	Vertical stretch, a>1 y= a\|x\|
Vertical Stretch or Shrink	Vertical shrink, 0< a< 1 y= a\|x\|

Transformations of y=\|x\|
Reflections	In the x-axis y=- \|x\|
Reflections	In the y-axis y=\|- x\|

Transformations of Absolute Value Functions

Catch-Up and Review

Investigating the Translations of Absolute Value Functions

Translations of Absolute Value Functions

Matching Graphs with Their Functions

Hint

Solution

Identifying and Correcting Error

Hint

Solution

Identifying the Translations of an Absolute Value Graph

Investigating the Transformations of an Absolute Value Function

Stretch and Shrink of a Function

Vertical Stretch and Shrink of a Function

Horizontal Stretch and Shrink of a Function

Stretch and Shrink of an Absolute Value Function

Draining and Refilling Water Tank

Hint

Solution

Reflection of a Function

Reflection of an Absolute Value Function

Construction via Reflection

Hint

Solution

Constructing an Absolute Value Function from a Linear Function

Transformations of Absolute Value Functions

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