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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.

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.

By adding some number to every function value, $g(x)=f(x)+k,$ 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. $g(x)=f(x−h)$ Note that the number, $h,$ is subtracted and not added. This is so that a positive $h$ leads to a translation to the right, which is the positive $x$-direction.

Translate graph to the right

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

Reflect graph in $x$-axis

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

Reflect graph in $y$-axis

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant $a>0$: $g(x)=a⋅f(x).$ All vertical distances from the graph to the $x$-axis are changed by the factor $a.$ Thus, preserving any $x$-intercepts.

Stretch graph vertically

By instead multiplying the input of a function rule by some constant $a>0,$ $g(x)=f(a⋅x),$ its graph will be horizontally stretched or shrunk by the factor $a1 .$ Since the $x$-value of $y$-intercepts is $0,$ they are not affected by this transformation.

Stretch graph horizontally

The rules of $f$ and $g$ are given such that $g$ is a transformation of $f.$ $f(x)=∣x−2∣g(x)=f(x+1)−3$ Describe the transformation(s) $f$ underwent to become $g.$ Then, state the $y$-intercept of $g.$

Show Solution

To begin, we'll analyze the given function rules. Adding $1$ to the input of $f,$ and then subtracting the output by $3,$ gives the function $g.$ 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 $1$ unit.

We've established that $f(x+1)$ is a translation of $f(x)$ to the left by $1$ unit. Now, we can view $g(x)=f(x+1)−3$ as a downward translation of $f(x+1)$ by $3$ units.

Thus, $f$ has been translated $1$ unit to the left **and** $3$ units downward to become $g.$ The $y$-intercept is where the graph crosses the $y$-axis, at which point the $x$-coordinate is $0.$ We'll find the $y$-coordinate by substituting $x=0$ into the rule of $g(x).$

To fully evaluate $g(0),$ we first have to find $f(1)$ by substituting $x=1$ into the rule of $f(x).$

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

Thus, the $y$-intercept of $g$ is $(0,-2).$

Below, the graphs of $f,$ $g,$ and $h$ are shown. Find the rules of $g$ and $h,$ expressed as transformations of $f.$

Show Solution

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

By comparing their function values at different $x$-values, we can find the factor by which it's been shrunk. For instance, choosing $x=-3$ and $x=3$ gives us the information we need.

At $x=-3,$ the function values are $4$ and $2.$ At $x=3,$ the function values are $2$ and $1.$ Thus, we can conclude that every function value of $g$ is half of that of $f.$ This transformation is written algebraically as $g(x)=0.5f(x).$ Comparing $f$ with $h,$ we can see that the slope of their respective piecewise-linear parts are of different signs. When $f$ increases, $g$ decreases, and vice versa. The only transformation that creates this effect is a reflection in the $x$-axis. However, simply reflecting $f$ in the $x$-axis doesn't result in $h.$

As we can see from the graph, $f$ has to be translated $2$ units downward after the reflection to match $h.$ In function notation, this is written as $h(x)=-f(x)−2,$ where $-f(x)$ is the reflection, and then subtracting by $2$ translates the graph downward $2$ units.

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