2.3 Vertical Transformers
Transformers
A transformer is a function that is used to produce geometric effects on another function's graph. For any constant \( c : \Number \), we have a constant-adder transformer
and a constant-multiplier transformer.
The negation transformer is the constant-multiplier that multiplies by negative one.
We'll look at using transformers on a function's output in the remainder of this section, and we'll look at using transformers on a function's input in the next.
Vertical Transformers
Applying a transformer to the output of a function produces a vertical geometric effect on the graph. The following schematic depicts a vertical transformation!
Suppose \( z \depends z^{\prime} \) is a transformer, and \( z^{\prime} \depends x \) is a function with a known graph. Then we can understand the graph of the function \( z \depends x \) according to the following table.
| algebraic relationship | geometric relationship |
|---|---|
| \( z = z^{\prime} + 3 \) | translate up by three |
| \( z = z^{\prime} - 3 \) | translate down by three |
| \( z = 2 z^{\prime} \) | vertically stretch by a factor of two |
| \( z = z^{\prime} / 2 \) | vertically shrink by a factor of two |
| \( z = - z^{\prime} \) | vertically reflect |
Stretching, shrinking, and reflecting happen with respect to the \( x \)-axis.
Example.
Let's graph the function
\[
\begin{align}
z &\mathrel{\overset{f}{\leftarrow}} x
\\
z &= 2 \mult \sqrt[2]{x}.
\end{align}
\]
We'll start by drawing a schematic for \( f \).
By introducing the variable \( z^{\prime} \) on the intermediate wire, we can decompose \( f \).
\[
\begin{align}
z &\depends z^{\prime}
&
&
&
z^{\prime} &\depends x
\\
z &= 2 \mult z^{\prime}
&
&\andSpaced
&
z^{\prime} &= \sqrt[2]{x}
\end{align}
\]
We know how to graph the square root function, \( z^{\prime} \depends x \). To graph the function \( z \depends x \), each height \( z \) is twice the height \( z^{\prime} \).
