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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.

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 | $g(x)=f(x)+k$ |

Horizontal Translation | $g(x)=f(x−h)$ |

Reflection in the $x-$axis | $g(x)=-f(x)$ |

Reflection in the $y-$axis | $g(x)=f(-x)$ |

Vertical Stretch and Shrink | $g(x)=a⋅f(x)$ |

Horizontal Stretch and Shrink | $g(x)=f(a⋅x)$ |

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

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)=-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.

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

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

Show 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 $x−3.$ $f(x)=2x+4⇔f(x−3)=2(x−3)+4$ 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.

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.

A function graph can be vertically stretched or shrunk by multiplying the function rule by some factor $a>0.$ Algebraically, $g(x)=a⋅f(x).$ 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 $0.$ Thus, they are not affected by this transformation.

Stretch graph vertically

By multiplying the input of a function by a factor $a>0,$ its graph can be horizontally stretched or shrunk. $g(x)=f(a⋅x)$ 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 $a1 .$

Stretch graph horizontally

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

Show 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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