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Describing Transformations of Absolute Value Functions

Similar to linear functions, absolute value functions can be transformed to create other absolute value functions. Below, transformations of absolute value functions will be explored.
Rule

Transformations of Absolute Value Functions

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.

Rule

Translation

By adding some number to every function value, a function graph is translated vertically.

Translate graph upward

A graph is translated horizontally by subtracting a number from the input of the function rule. Note that the number, is subtracted and not added. This is so that a positive leads to a translation to the right, which is the positive -direction.

Translate graph to the right

Rule

Reflection

A function is reflected in the -axis by changing the sign of all function values: Graphically, all points on the graph move to the opposite side of the -axis, while maintaining their distance to the -axis.

Reflect graph in -axis

A graph is instead reflected in the -axis, by moving all points on the graph to the opposite side of the -axis. This occurs by changing the sign of the input of the function. Notice that the vertex of the graph changes location when it does not lie on the line of reflection.

Reflect graph in -axis

Rule

Stretch and Shrink

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant : All vertical distances from the graph to the -axis are changed by the factor Thus, preserving any -intercepts.

Stretch graph vertically

By instead multiplying the input of a function rule by some constant its graph will be horizontally stretched or shrunk by the factor Since the -value of -intercepts is they are not affected by this transformation.

Stretch graph horizontally

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Exercise

The rules of and are given such that is a transformation of Describe the transformation(s) underwent to become Then, state the -intercept of

Show Solution
Solution

To begin, we'll analyze the given function rules. Adding to the input of and then subtracting the output by gives the function We can recognize the addition to the input as a horizontal translation, but in which direction? When the input of a function is increased by some number the graph is translated to the left. Thus, this is a translation to the left, by unit.

We've established that is a translation of to the left by unit. Now, we can view as a downward translation of by units.

Thus, has been translated unit to the left and units downward to become The -intercept is where the graph crosses the -axis, at which point the -coordinate is We'll find the -coordinate by substituting into the rule of

To fully evaluate we first have to find by substituting into the rule of

We can now use this value to find the -coordinate.

Thus, the -intercept of is

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Exercise

Below, the graphs of and are shown. Find the rules of and expressed as transformations of

Show Solution
Solution

We'll start by comparing the graphs of and Their vertices are at the same point, and is flatter than This flattening could either be seen as the graph being stretched horizontally, or shrunk vertically, but which is it? The -intercept has been affected, which rules out the horizontal stretch. At the same time, the -intercept has been preserved, which confirms the vertical shrink.

By comparing their function values at different -values, we can find the factor by which it's been shrunk. For instance, choosing and gives us the information we need.

At the function values are and At the function values are and Thus, we can conclude that every function value of is half of that of This transformation is written algebraically as Comparing with we can see that the slope of their respective piecewise-linear parts are of different signs. When increases, decreases, and vice versa. The only transformation that creates this effect is a reflection in the -axis. However, simply reflecting in the -axis doesn't result in

As we can see from the graph, has to be translated units downward after the reflection to match In function notation, this is written as where is the reflection, and then subtracting by translates the graph downward units.

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