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Linear Functions

Describing Transformations of Linear Functions

Concept

Function Family

Functions of similar characteristics can be grouped together in what is known as a function family. By applying one or several transformations to a parent function, it is possible to construct any function in its function family. The most common function families are linear functions, absolute value functions, quadratic functions, and exponential functions.

Linear, absolute value, quadratic, exponential functions

There are also several other function families such as square root functions, cubic functions, rational functions, and logarithmic functions.

square root, cubic, rational, logarithmic functions

Theory

Types of Transformations

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

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

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 x- or y-axis. A reflection in the x-axis is achieved by changing the sign of the y-coordinate of every point on the graph. Algebraically, this is expressed as
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.
When x=0, which is at any y-intercept, this reflection doesn't affect the input value. Thus, y-intercepts are preserved by reflections in the y-axis.
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Exercise
The rules of f and h are given such that h is a transformation of f.
Describe the transformation(s) f underwent to become h. Then, state the slope and the y-intercept of h.
Show Solution
Solution

To begin, we'll analyze the given function rules. Subtracting the input of f by 3 gives the function h. We can recognize this as a horizontal translation, but in which direction? When the input of a function is reduced by some number, 3 in this case, greater input values are needed to achieve the same output values. Thus, this is a translation to the right, by 3 units.

From the graph above, it's possible to find the slope and y-intercept. However, this is not a reliable method, since the graph is only supposed to be a sketch. Instead, we can find them algebraically. h is defined as
h(x)=f(x3),
which means that the rule of h can be found by reducing the input in the rule of f by 3. In practice, this is done by replacing every x in the rule of f with x3.
Substituting this into the rule for h gives
h(x)=2(x3)+4.
Simplifying the right-hand side will lead to h being expressed in slope-intercept form, allowing us to identify its slope and y-intercept.
h(x)=2(x3)+4
h(x)=2x6+4
h(x)=2x2

We can now identify the slope, 2, and the y-intercept, (0,-2). 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 a>0. Algebraically,
g(x)=af(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. 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 a>0, its graph can be horizontally stretched or shrunk.
g(x)=f(ax)
If a>1, every input value will be changed as though it was further away from x=0 than it really is. This leads to the graph being shrunk horizontally — every part of the graph is moved closer to the y-axis. In the same fashion, a<1 leads to a horizontal stretch. The horizontal distance between the graph and the y-axis is changed by a factor of
Stretch graph horizontally

y-intercepts have the x-value 0, which is why they are not affected by this transformation.
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Exercise

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

Show Solution
Solution
From the graph we can see that f(x) and g(x) have slopes of different signs. This is an indicator that g is some reflection of f, but in what line was it reflected? The x-intercept, (2,0), is preserved by this transformation, which is agreeable with a reflection in the x-axis:
g(x)=-f(x).
To confirm whether this is the case, we can choose a few x-values to find their corresponding y-values. If the functions have values of different signs at every x, then g must be a reflection of f in the x-axis. Choosing x=3, we find the function values f(3)=1 and g(3)=-1, which have different signs.
Looking at x=-1, we find that f(-1)=-3 and g(-1)=3, also different signs. Choosing any other x-value would yield the same result. Thus, we know that g is a reflection of f in the x-axis, and can be expressed as
g(x)=-f(x).
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