# 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.
Rule

## 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 $g(x)=f(x){\color{#0000FF}{\vphantom{ }+k}}$
Horizontal Translation $g(x)=f(x{\color{#0000FF}{\vphantom{ }-h}})$
Reflection in the $x$-axis $g(x)={\color{#0000FF}{\text{-}}}f(x)$
Reflection in the $y$-axis $g(x)=f({\color{#0000FF}{\text{-} }}x)$
Vertical Stretch and Shrink $g(x)={\color{#0000FF}{a \cdot\vphantom{ }}}f(x)$
Horizontal Stretch and Shrink $g(x)=f({\color{#0000FF}{a \cdot\vphantom{ }}}x)$
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 $g(x) = f(x) + k,$ where $k$ is some number defining the vertical translation, and $g(x)$ is the translated function. A positive $k$ increases the value of the function for every $x,$ moving the graph upward. Similarly, a negative $k$ 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 $g(x) = f(x - h),$ where $h$ is some number giving the horizontal translation. A positive $h$ reduces the input value, $x - h,$ making it as though every $x$-value is smaller than it really is. Thus, greater $x$-values are needed to get the same result, leading to a translation to the right. Similarly, a negative $h$ gives a translation to the left.

Translate graph to the right

Rule

## Reflection

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 $g(x) = \text{-} f(x).$ 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.

Reflect graph in $x$-axis

A reflection in the $y$-axis is instead achieved by changing the sign of every input value. $g(x) = f(\text{-} x)$ 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.

Reflect graph in $y$-axis

Exercise

The rules of $f$ and $h$ are given such that $h$ is a transformation of $f.$ $f(x)=2x+4 \qquad h(x)=f(x-3)$ Describe the transformation(s) $f$ underwent to become $h.$ Then, state the slope and the $y$-intercept of $h.$

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 $h(x) = f(x - 3),$ 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 ${\color{#0000FF}{x - 3}}.$ $f(x) = 2x + 4 \quad \Leftrightarrow \quad f({\color{#0000FF}{x - 3}}) = 2({\color{#0000FF}{x - 3}}) + 4$ Substituting this into the rule for $h$ gives $h(x) = 2(x - 3) + 4.$ Simplifying the right-hand side will lead to $h$ being expressed in slope-intercept form, allowing us to identify its slope and $y$-intercept.

$h(x) = 2(x - 3) + 4$
$h(x) = 2x - 6 + 4$
$h(x) = 2x - 2$

We can now identify the slope, $2,$ and the $y$-intercept, $(0, \text{-} 2).$ 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.

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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, $g(x) = a \cdot f(x).$ 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 $0.$ 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. $g(x) = f(a \cdot x)$ 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 $\frac 1 a.$

Stretch graph horizontally

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

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

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: $g(x) = \text{-} f(x).$ 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 $g(3) = \text{-} 1,$ which have different signs.

Looking at $x = \text{-} 1,$ we find that $f(\text{-} 1) = \text{-} 3$ and $g(\text{-} 1) = 3,$ 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 $g(x) = \text{-} f(x).$

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