To match each function with its respective period, follow these steps:
Step 1: Identify the periods of each function based on the given results:
1. \( y = 5 \cot \pi - 8 \) has a period of \(\pi\).
2. \( y = \frac{2}{5} \sin \left( \frac{2}{z} - \pi \right) \) has a period of \(2\pi\).
3. \( y = -\frac{1}{3} \csc 2x \) has a period of \(\pi\).
4. \( y = 5 \sec (2x + 2\pi) \) has a period of \(\pi\).
5. \( y = -6 \cot x - 10 \) has a period of \(\pi\).
6. \( y = -3 \tan \frac{x}{2} \) has a period of \(4\pi\).
Step 2: Place each function into the appropriate category on the chart.
### Chart:
[tex]\[
\begin{array}{|l|l|}
\hline
\pi & 2\pi \\
\hline
y = 5 \cot \pi - 8 & y = \frac{2}{5} \sin \left( \frac{2}{z} - \pi \right) \\
y = -\frac{1}{3} \csc 2x & \\
y = 5 \sec (2x + 2\pi) & \\
y = -6 \cot x - 10 & \\
& \\
\hline
\end{array}
\][/tex]
### Final Result:
[tex]\[
\begin{array}{|l|l|}
\hline
\pi & 2\pi \\
\hline
y = 5 \cot \pi - 8 & y = \frac{2}{5} \sin \left( \frac{2}{z} - \pi \right) \\
y = -\frac{1}{3} \csc 2x & \\
y = 5 \sec (2x + 2\pi) & \\
y = -6 \cot x - 10 & \\
& \\
\hline
\end{array}
\][/tex]
In summary, the functions have been matched with their respective periods as follows:
- \(\pi\):
- \( y = 5 \cot \pi - 8 \)
- \( y = -\frac{1}{3} \csc 2 x \)
- \( y = 5 \sec (2 x + 2 \pi) \)
- \( y = -6 \cot x - 10 \)
- \(2\pi\):
- \( y = \frac{2}{5} \sin \left( \frac{2}{z} - \pi \right) \)
The function [tex]\( y = -3 \tan \frac{x}{2} \)[/tex] was assigned a period of [tex]\(4\pi\)[/tex] earlier and is not included in the [tex]\(\pi\)[/tex] or [tex]\(2\pi\)[/tex] columns.
