2.4 Horizontal Transformers
Transforming the input of a function produces a horizontal geometric effect. Here's a schematic depicting a horizontal transformation.
Suppose \( z \depends x^{\prime} \) is a function with a known graph, and \( x^{\prime} \depends x \) is a transformer. Then we can understand the graph of the function \( z \depends x \).
| algebraic relationship | geometric relationship |
|---|---|
| \( x = x^{\prime} + 3 \) | translate right by three |
| \( x = x^{\prime} - 3 \) | translate left by three |
| \( x = 2 x^{\prime} \) | horizontally stretch by a factor of two |
| \( x = x^{\prime} / 2 \) | horizontally shrink by a factor of two |
| \( x = - x^{\prime} \) | horizontally reflect |
Horizontal transformers looks a lot like vertical transformers, but there is an important subtle distinction to working with inputs!
Important.
We apply a horizontal transformer in the form
\[
\begin{align}
x^{\prime} &\depends x,
\end{align}
\]
but we understand the geometry of a horizontal transformer in the form
\[
\begin{align}
x &\depends x^{\prime}.
\end{align}
\]
This is best understood with examples.
Example.
Consider the function
\[
\begin{align}
z &\mathrel{\overset{f}{\leftarrow}} x
\\
z &= (x - 2)^3.
\end{align}
\]
Let's draw a schematic to understand how to graph this function.
By placing the variable \( x^{\prime} \) on the intermediate wire, we can decompose \( f \) as a known function and a transformer.
\[
\begin{align}
z &\depends x^{\prime}
&
&
&
x^{\prime} &\depends x
\\
z &= {x^{\prime}}^3
&
&\andSpaced
&
x^{\prime} &= x - 2
\end{align}
\]
We'll make sense of this horizontal transformer by solving for \( x \).
\[
\begin{align}
x &\depends x^{\prime}
\\
x &= x^{\prime} + 2
\end{align}
\]
Each \( x \)-value is two more than the corresponding \( x^{\prime} \)-value. To graph \( f \), we translate the graph of the cubing function two to the right.
Let's also see an example that includes multiple horizontal transformers.
Example.
Consider the function
\[
\begin{align}
z &\mathrel{\overset{f}{\leftarrow}} x
\\
z &= \biggl(\frac{x + 1}{2}\biggr)^2.
\end{align}
\]
We'd like to graph this function by understanding its transformers. We'll start by drawing a schematic.
We can decompose \( f \) as a known function and transformers.
\[
\begin{align}
z &\depends x^{\prime}
&
&
&
x^{\prime} &\depends x
\\
z &= {x^{\prime}}^2
&
&\andSpaced
&
x^{\prime} &= \frac{x + 1}{2}
\end{align}
\]
Let's invert the transformers by solving for \( x \).
\[
\begin{align}
x &\depends x^{\prime}
\\
x &= 2 x^{\prime} - 1
\end{align}
\]
To produce an \( x \)-value, we double an \( x^{\prime} \)-value and then subtract one.
To graph \( f \), we take our known parabola, horizontally stretch by a factor of two, and then translate one to the left.