Practice with absolute value functions with transformation and guidance

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.

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Investigating the Translations of Absolute Value Functions

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.

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.

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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.

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.

Translation up k units , k > 0 y = ∣ x ∣ + k ​ Translation down k units , k < 0 y = ∣ x ∣ + k ​ Translation to the right h units , h > 0 y = ∣ x − h ∣ ​ Translation to the left h units , h < 0 y = ∣ x − h ∣ ​ info Info error Report error reply Share

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. <"type":"pair","form":<"alts":[[<"id":0,"text":"Graph A">,,,],[fxxfxxfxxfxxHint

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.

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

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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! ,"formTextBefore":"","formTextAfter":"","answer":1>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 f ( x − 5 ) = 2 ∣ x − 5 + 2 ∣ − 1 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 f ( x − 5 ) − 3 = 2 ∣ x − 3 ∣ − 1 − 3 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. info Info error Report error reply Share

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 .

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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.

Function I ​ y = a ( ∣ x − 1 ∣ − 1 ) ​ 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 .

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Stretch and Shrink of a Function

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 ⋅ 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.

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 shrink, 0 < a < 1
y = a f ( x )

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 ⋅ x ) ​

If 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 b 1 ​ .

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 shrink, b > 1
y = f ( b x )