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The functions belonging to the same function family can be transformed into each other by translations, stretching and shrinking, or reflections. By applying one or several transformations to a parent function, it is possible to obtain any function from its function family. This lesson will focus on the transformations of absolute value functions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Function Family
Absolute Value Functions
Function Notation
Parent Function
Transformations of Linear Functions

Explore

Investigating the Translations of Absolute Value Functions

Consider the graph of the parent absolute value function.

Graph of the parent function y = absolute value of x

Many different absolute value functions can be obtained by shifting the graph of the parent absolute value function. Absolute value functions obtained in this way have the following form.

y = ∣ x - h ∣ + k

In this equation,

h

and

k

are real numbers. Using the following applet, investigate how the values of

h

and

k

affect the graph of the parent function.

Interactive graph that allows to explore the effect of the parameters h and k

The same types of transformations that create new linear functions also do the same for absolute value functions. They affect absolute value functions in the same way, as well. However, since linear functions and absolute value functions have some significant differences, the transformations might look different graphically.

Discussion

Translations of Absolute Value Functions

The graph of an absolute value function

y = ∣ x ∣

can be translated vertically by adding a number to — or subtracting from — the function rule.

Vertical Translation of Absolute Value Function

Likewise, it can be also translated horizontally by adding a number to — or subtracting from — the rule's input.

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

Example

Matching Graphs with Their Functions

Tadeo just learned about translations of absolute value functions. He believes in the motto that practice makes perfect, so he decides to study more. The following graphs are the graphs of the absolute value parent function after a certain translation.

Help Tadeo match each graph with the corresponding function rule.

Hint

Compare the given graphs to the graph of the absolute value parent function to identify the translation applied to each graph.

Solution

Begin by identifying the translation of each graph when compared to the graph of the absolute value parent function $f (x) = ∣ x ∣ .$

Now that the translations have been identified, recall the translation rules.

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

Using this table, the function rules of the graphs can be written.

Graph A : Graph B : Graph C : Graph D : f (x) = ∣ x ∣ - 3 f (x) = ∣ x + 2 ∣ f (x) = ∣ x - 4 ∣ f (x) = ∣ x ∣ + 1

Example

Identifying and Correcting Error

Tadeo and Emily are classmates in North High School. They have been asked to translate the following absolute value function

5

units to the right and then

3

units down.

f (x) = 2 ∣ x + 2 ∣ - 1

Yet, they obtained different results.

By performing the required translations, determine who is correct!

Hint

The graph of an absolute value function $y = ∣ x ∣$ can be translated vertically by adding a number to — or subtracting from — the function rule. Likewise, it can be also translated horizontally by adding a number to — or subtracting from — the rule's input.

Solution

An absolute value function can be translated

5

units to the right by subtracting

5

from the function rule's input. To do so, substitute

x - 5

for

x

into

f (x) = 2 ∣ x + 2 ∣ - 1 .

f (x) = 2 ∣ x + 2 ∣ - 1

Substitute

$x = x - 5$

f (x - 5) = 2 ∣ x - 5 + 2 ∣ - 1

AddTerms

Add terms

f (x - 5) = 2 ∣ x - 3 ∣ - 1

Next, to translate the resulting function

3

units down,

3

must be subtracted from the function rule.

f (x - 5) = 2 ∣ x - 3 ∣ - 1

SubEqn

$LHS - 3 = RHS - 3$

f (x - 5) - 3 = 2 ∣ x - 3 ∣ - 1 - 3

SubTerm

Subtract term

f (x - 5) - 3 = 2 ∣ x - 3 ∣ - 4

Finally, the result can be simplified by replacing

f (x - 5) - 3

with

g (x) .

g (x) = 2 ∣ x - 3 ∣ - 4

As a result, Emily is correct.

Pop Quiz

Identifying the Translations of an Absolute Value Graph

The following applet shows the graph of an absolute value function in the form of $f (x) = ∣ x - h ∣ + k,$ where $h$ and $k$ are integers. Considering the translation rules, determine the values of $h$ and $k .$

Interactive graph showing different translations of the absolute value parent function

Explore

Investigating the Transformations of an Absolute Value Function

Apart from translations, new absolute value functions can be constructed by shrinking or stretching an absolute value function. Consider the following functions.

\underline{Function I} y = a (∣ x - 1 ∣ - 1) \underline{Function II} y = ∣ b x - 1 ∣ - 1

In these examples,

a

and

b

are real numbers greater than

0 .

Investigate how the values of

a

and

b

change the graph of

y = ∣ x - 1 ∣ - 1 .

Interactive graph showing the effects of the parameters a and b

Discussion

Stretch and Shrink of a Function

Concept

Vertical Stretch and Shrink of a Function

The graph of a function can be vertically stretched or shrunk by multiplying the function rule by a positive number

a .

y = a \cdot f (x)

The vertical distance between the graph and the

x -

axis will then change by the factor

a

at every point on the graph. If

a > 1,

this will lead to the graph being stretched vertically. Similarly,

a < 1

leads to the graph being shrunk vertically. Note that

x -

intercepts have the function value

0 .

Therefore, they are not affected by this transformation.

Stretching and Shrinking of a Graph Vertically

The general form of this transformation is shown in the table.

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

Concept

Horizontal Stretch and Shrink of a Function

By multiplying the input of a function by a positive number

b,

its graph can be horizontally stretched or shrunk.

y = f (b \cdot x)

b > 1,

every input value will be changed as though it was farther away from the

y -

axis than it really is. This leads to the graph being shrunk horizontally — every part of the graph is moved closer to the

y -

axis. Conversely,

b < 1

leads to a horizontal stretch. The horizontal distance between the graph and the

y -

axis is changed by a factor of

\frac{1}{b} .

Stretching and Shrinking of a Graph Horizontally

Note that

y

-intercepts have the

x

-value

0,

which is why they are not affected by this transformation. The general form of this transformation is shown in the table.

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

Discussion

Stretch and Shrink of an Absolute Value Function

A stretch and shrink of an absolute value function is a transformation that changes the width of the graph of the function without shifting it. The graph of an absolute value function

y = ∣ x - 1 ∣ - 1

can be stretched or shrunk vertically by multiplying the function rule by a positive number.

Vertical Stretch and Shrink of Absolute Value Function

Likewise, the graph of an absolute value function

y = ∣ x - 0.5 ∣

can be also stretched or shrunk horizontally by multiplying the rule's input by a positive number.

Horizontal Stretch and Shrink of Absolute Value Function

The table below summarizes the different types of stretch and shrink that can be performed for an absolute value function.

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

Example

Draining and Refilling Water Tank

Emily uses a water tank that contains $10$ cubic meters of water to water her plants. She drains the water tank from one faucet and then refills it from another identical faucet.

The following graph models the water level of the tank when it is drained and then refilled after $t$ minutes.

Graphs of the absolute value function f(t)= 1/2 |t-20|

a Emily notices that when she doubles both the amount of the water and the diameter of the faucets, the graph of the function vertically stretches by a factor of

2 .

If that is the case, what is the function of the transformed graph.

b If Emily only doubles the diameter of the faucets, the graph of the function shrinks horizontally. In this case, what would be the function of the new graph?

Hint

a The graph of an absolute value function

y = f (x)

can be stretched or shrunk vertically by multiplying the function rule by a positive number.

b The graph of an absolute value function

y = f (x)

can be stretched or shrunk horizontally by multiplying the rule's input by a positive number.

Solution

a Recall that, when Emily doubles both the amount of the water and the diameter of the faucets, the graph of the function vertically stretches by a factor of

2 .

To stretch the graph of an absolute value function vertically, the function rule should be multiplied by a positive number. In this case, it should be multiplied by

2 .

With this information, the function of the transformed graph can be written.

f (t) = \frac{1}{2} ∣ t - 20 ∣ ⇓ f (t) = ∣ t - 20 ∣

b This time Emily doubles only the diameter of the faucets. As a result the the graph of the function shrinks horizontally by factor of

2 .

To shrink the graph of an absolute value function horizontally, the function rule's input should be multiplied by a positive number. In this case, it should be multiplied by

2 .

f (t) = \frac{1}{2} ∣ t - 20 ∣ ⇓ f (t) = \frac{1}{2} ∣ 2 t - 20 ∣

Discussion

Reflection of a Function

A reflection of a function is a transformation that flips a graph over some line. This line is called the line of reflection and is commonly either the

x -

y -

axis. A reflection in the

x -

axis is achieved by changing the sign of the

y -

coordinate of every point on the graph.

y = - f (x)

The

y -

coordinate of all

x -

intercepts is

0 .

Thus, changing the sign of the function value at

x -

intercepts makes no difference — any

x -

intercepts are preserved when a graph is reflected in the

x -

axis.

A reflection in the

y -

axis is instead achieved by changing the sign of every input value.

y = f (- x)

When

x = 0,

which is at the

y -

intercept, this reflection does not affect the input value. Therefore, the

y -

intercept is preserved by reflections in the

y -

axis.

The following table illustrates the different types of reflections that can be done to a function.

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

Discussion

Reflection of an Absolute Value Function

One last transformation that can be applied to absolute value functions are reflections. A reflection of an absolute value function is a transformation that flips the graph without changing its shape. The graph of an absolute value function

y = ∣ x - 1 ∣ - 1

can be reflected in the

x -

axis by multiplying the function rule by

- 1 .

Reflection of an Absolute Value Function in the x-axis

In a similar way, the graph of an absolute value function

y = ∣ 2 x - 2 ∣ - 1

can be also reflected in the

y -

axis by multiplying the rule's input by

- 1 .

Reflection of an Absolute Value Function in the y-axis

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

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

Example

Construction via Reflection

a As a result of reflecting an absolute value function, some symmetric figures can be obtained. Currently, Emily is working on the following absolute value function.

f (x) = ∣ 2 x - 5 ∣

By reflecting this function, she wants to obtain the letter W. Write the equation of the reflected function by determining the type of the reflection.

b By reflecting an absolute value function, a quadrilateral can also be obtained. Consider the following absolute value function.

f (x) = \frac{1}{2} ∣ x - 4 ∣ - 2

How should Emily reflect this function to get a quadrilateral? What will be the equation of the reflected function?

Hint

a Begin by graphing the given function. Note that the graph of an absolute value function

y = f (x)

can be reflected in the

x -

axis by multiplying the function rule by

- 1 .

In a similar way, the function can be also reflected in the

y -

axis by multiplying the rule's input by

- 1 .

b Begin by graphing the given function. Note that the graph of an absolute value function

y = f (x)

can be reflected in the

x -

axis by multiplying the function rule by

- 1 .

In a similar way, it can be also reflected in the

y -

axis by multiplying the rule's input by

- 1 .

Solution

a To see how to obtain the letter W, begin by graphing the given function by using a table of values.

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

Now plot the ordered pairs and connect them to graph the absolute value function.

Looking at the graph of the function, it can be concluded that

f (x)

needs to be reflected in the

y -

axis.

Recall that the graph of an absolute value function can be reflected in the

y -

axis by multiplying the function rule's input by

- 1 .

\underline{Given Function} f (x) = ∣ 2 x - 5 ∣ ⇓ \underline{Reflected Function} g (x) = ∣ - 2 x - 5 ∣

b In a similar way, draw the graph of

f (x)

to identify the type of the reflection.

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

Plot the ordered pairs and draw the graph.

It can be seen that, to form a quadrilateral, the graph of the function needs to be reflected in the

x -

axis.

Note that the graph of an absolute value function can be reflected in the

x -

axis by multiplying the function rule by

- 1 .

\underline{Given Function} f (x) = \frac{1}{2} ∣ x - 4 ∣ - 2 ⇓ \underline{Reflected Function} g (x) = - \frac{1}{2} ∣ x - 4 ∣ + 2

Closure

Constructing an Absolute Value Function from a Linear Function

Throughout the lesson, the transformations of absolute value functions have been covered. It has been seen that different absolute value functions can be obtained by transforming the absolute value parent function or another absolute value function. However, this is not the only way to obtain an absolute value function. Consider a linear function in slope-intercept form.

f (x) = \frac{1}{2} x - 1

Draw the graph of this function and reflect the negative part of the graph in the

x -

axis.

The resulting function is an absolute value function of the form

g (x) = ∣ m x + b ∣,

where

m

and

b

are real numbers and

m

cannot be

0 .

g (x) = ∣ ∣ ∣ ∣ ∣ \frac{1}{2} x - 1 ∣ ∣ ∣ ∣ ∣

Therefore, absolute value functions in the form

g (x) = ∣ m x + b ∣

can be obtained by reflecting the negative part of the graph of a linear function in the form

f (x) = m x + b .

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution