5.4 Five Graphs: Bend
We've analyzed our five graphs for height and for slope in previous chapters. Let's revisit these graphs one more time, this time looking at the bend!
Squaring
As we just discussed in the preceding section, the squaring function,
\[
\begin{align}
z &\depends x
\\
z &= x^2,
\end{align}
\]
has a constant bend of one.
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= 1
\end{align}
\]
The graph is concave up everywhere.
Cubing
Now let's analyze the cubing function,
\[
\begin{align}
z &\depends x
\\
z &= x^3.
\end{align}
\]
The cubing function has bend
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= 3 x.
\end{align}
\]
The bend is positive for positive \( x \)-values, is negative for negative \( x \)-values, and is zero at \( x = 0 \). The cubing function illustrates something that is rather rare! Not only is \( x = 0 \) a level point, but it is also an inflection point.
Reciprocal
Let's take a look at the reciprocal function,
\[
\begin{align}
z &\depends x
\\
z &= \frac{1}{x}.
\end{align}
\]
We can calculate the bend.
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= \frac{1}{x^3}
\end{align}
\]
Although the bend transitions \( x = 0 \), we won't call this an inflection point! We'll want the bend to be defined at inflection points.
Square Root
Let's consider the square root function,
\[
\begin{align}
z &\depends x
\\
z &= \sqrt[2]{x}.
\end{align}
\]
This function has bend
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= \frac{-1}{8 x \mult \sqrt[2]{x}}.
\end{align}
\]
The graph is concave down at all allowed inputs, \( x \gt 0 \).
Cube Root
Last but not least, let's look at the graph of the cube root,
\[
\begin{align}
z &\depends x
\\
z &= \sqrt[3]{x}.
\end{align}
\]
We can calculate the bend to be
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= \frac{-1}{9 \mult \bigl(\sqrt[3]{x}\bigr)^5}.
\end{align}
\]
The graph is concave down for positive \( x \)-values and concave up for negative \( x \)-values.
