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Linear Functions

Describing Transformations of Linear Functions

Concept

Function Families

Functions of similar characteristics can be grouped together in what is known as a function family. One such family is the linear functions, which consists of every function with a constant rate of change. By applying one or several transformations to a parent function, it is possible to construct any function in its function family.

Theory

Types of Transformations

The table below shows different types of transformations a function can undergo. Notice the corresponding function notation, where is the original function and is the transformed function.

Transformation Function Notation
Vertical Translation
Horizontal Translation
Reflection in the axis
Reflection in the axis
Vertical Stretch and Shrink
Horizontal Stretch and Shrink

Rule

Translation

A translation is a transformation that moves a graph in some direction, without any rotation, shrinking, stretching, etc. A function's graph is vertically translated by adding some number to the function rule. Generally, this is written as where is some number defining the vertical translation, and is the translated function. A positive increases the value of the function for every moving the graph upward. Similarly, a negative moves the graph downward.

Translate graph upward

A function's graph is horizontally translated by instead subtracting some number from the input of the function rule. In function notation, this is written as where is some number giving the horizontal translation. A positive reduces the input value, making it as though every -value is smaller than it really is. Thus, greater -values are needed to get the same result, leading to a translation to the right. Similarly, a negative gives a translation to the left.

Translate graph to the right

Concept

Reflection of a Function

A reflection is a transformation that flips a graph over some line. This line is called the line of reflection, and is commonly either the - or -axis. A reflection in the -axis is achieved by changing the sign of the -coordinate of every point on the graph. Algebraically, this is expressed as The -coordinate of all -intercepts is Thus, changing the sign of the function value at -intercepts makes no difference — any -intercepts are preserved when a graph is reflected in the -axis.

A reflection in the -axis is instead achieved by changing the sign of every input value. When which is at any -intercept, this reflection doesn't affect the input value. Thus, -intercepts are preserved by reflections in the -axis.

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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 slope and the -intercept of

Show Solution
Solution

To begin, we'll analyze the given function rules. Subtracting the input of by gives the function We can recognize this as a horizontal translation, but in which direction? When the input of a function is reduced by some number, in this case, greater input values are needed to achieve the same output values. Thus, this is a translation to the right, by units.

From the graph above, it's possible to find the slope and -intercept. However, this is not a reliable method, since the graph is only supposed to be a sketch. Instead, we can find them algebraically. is defined as which means that the rule of can be found by reducing the input in the rule of by In practice, this is done by replacing every in the rule of with Substituting this into the rule for gives Simplifying the right-hand side will lead to being expressed in slope-intercept form, allowing us to identify its slope and -intercept.

We can now identify the slope, and the -intercept, Notice that the slope wasn't affected by the translation. Since translations do nothing but move graphs, all translations preserve the slope of a linear function.

Rule

Vertical Stretch and Shrink

A function graph can be vertically stretched or shrunk by multiplying the function rule by some factor Algebraically, The vertical distance between the graph and the -axis will then change by the factor at every point on the graph. If this will lead to the graph being stretched vertically. Similarly, leads to the graph being shrunk vertically. Note that -intercepts have the function value Thus, they are not affected by this transformation.

Stretch graph vertically

Rule

Horizontal Stretch and Shrink

By multiplying the input of a function by a factor its graph can be horizontally stretched or shrunk. If every input value will be changed as though it was further away from than it really is. This leads to the graph being shrunk horizontally — every part of the graph is moved closer to the -axis. In the same fashion, leads to a horizontal stretch. The horizontal distance between the graph and the -axis is changed by a factor of

Stretch graph horizontally

-intercepts have the -value which is why they are not affected by this transformation.
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Exercise

The graph below shows two linear functions and Write the rule for as a transformation of

Show Solution
Solution

From the graph we can see that and have slopes of different signs. This is an indicator that is some reflection of but in what line was it reflected? The -intercept, is preserved by this transformation, which is agreeable with a reflection in the -axis: To confirm whether this is the case, we can choose a few -values to find their corresponding -values. If the functions have values of different signs at every then must be a reflection of in the -axis. Choosing we find the function values and which have different signs.

Looking at we find that and also different signs. Choosing any other -value would yield the same result. Thus, we know that is a reflection of in the -axis, and can be expressed as

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