2.2 Five Graphs: Height
Let's take a look at five functions that will make many appearances throughout this text. We'll review their graphs and get some practice drawing their transition diagrams.
Squaring
First let's consider the squaring function,
\[
\begin{align}
z &\depends x
\\
z &= x^2.
\end{align}
\]
The graph of this function is, no doubt, quite familiar: a parabola. Although the height is zero at \( x = 0 \), the height does not transition.
Cubing
Next we'll look at the cubing function,
\[
\begin{align}
z &\depends x
\\
z &= x^3.
\end{align}
\]
We can plot some points to produce a graph. Again we find height zero at \( x = 0 \), and this time the height does transition.
Reciprocal
Let's take a look at the reciprocal function,
\[
\begin{align}
z &\depends x
\\
z &= \frac{1}{x}.
\end{align}
\]
The graph of the reciprocal function is called a hyperbola. To avoid dividing by zero, we must disallow \( x = 0 \) as input to the reciprocal. We see that the height transitions at \( x = 0 \).
Square Root
The square root function,
\[
\begin{align}
z &\depends x
\\
z &= \sqrt[2]{x},
\end{align}
\]
is defined for positive inputs, \( x \ge 0 \), only. The height is zero at \( x = 0 \) and is positive for positive inputs.
Cube Root
The cube root function,
\[
\begin{align}
z &\depends x
\\
z &= \sqrt[3]{x},
\end{align}
\]
is defined for all inputs. The height transitions at \( x = 0 \).
