1. **State the problem:** We are given the function $y = \csc x + 3$ and want to understand its behavior. 2. **Recall the definition:** The cosecant function is defined as $\csc x = \frac{1}{\sin x}$. 3. **Rewrite the function:** $$y = \frac{1}{\sin x} + 3$$ 4. **Important notes:** - $\csc x$ is undefined where $\sin x = 0$, i.e., at $x = k\pi$ for any integer $k$. - The function has vertical asymptotes at these points. 5. **Behavior:** - As $x$ approaches $k\pi$ from the right or left, $\csc x$ tends to $\pm \infty$, so $y$ also tends to $\pm \infty$. - The graph of $y$ is the graph of $\csc x$ shifted upward by 3 units. 6. **Summary:** The function $y = \csc x + 3$ is the cosecant function shifted vertically up by 3, with vertical asymptotes at $x = k\pi$ where $k$ is any integer.