MTH 162 Test 1 Study Guide – Trigonometric Functions, Graphs, and Applications

Q4. Graph the function $y = -\csc x$

Background

Topic: Graphs of Trigonometric Functions

This question tests your understanding of how to graph the cosecant function, specifically $y = -\csc x$, including its asymptotes, period, and amplitude. The cosecant function is the reciprocal of the sine function, and the negative sign reflects the graph across the x-axis.

Key Terms and Formulas

• Cosecant Function: $\csc x = \frac{1}{\sin x}$

• Period: $2\pi$ (same as sine)

• Vertical Asymptotes: Occur where $\sin x = 0$ (i.e., $x = n\pi$, where $n$ is an integer)

• Reflection: The negative sign means the graph is reflected over the x-axis compared to $\csc x$.

Step-by-Step Guidance

1. Start by recalling the graph of $y = \sin x$. Identify the x-values where $\sin x = 0$ (these will be the vertical asymptotes for $y = -\csc x$).

2. Plot the vertical asymptotes at $x = n\pi$ for all integers $n$.

3. For each interval between asymptotes, sketch the reciprocal of the sine curve, but reflected over the x-axis (since the function is $-\csc x$).

4. Remember, the minimum and maximum points of $\sin x$ become the maximum and minimum points of $-\csc x$, but with opposite sign.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $y = -\csc x$ consists of branches opening downward between each pair of vertical asymptotes at $x = n\pi$. The graph never crosses the x-axis and is the reflection of $y = \csc x$ over the x-axis.