Question: Graph the following function: y = -3 + 2cot(\frac{\pi}{2} x + \frac{3\pi}{4}) Step 1 of 2: Identify the shape of the more basic function that has been shifted, reflected, stretched or compressed. Answer Graph shape: cotangent function, with vertical asymptotes and decreasing branches; position_hint: top-right.

1. **State the problem:** We need to analyze the function $$y = -3 + 2\cot\left(\frac{\pi}{2} x + \frac{3\pi}{4}\right)$$ and identify the shape of its basic graph. 2. **Recall the basic function:** The basic function here is the cotangent function $$\cot(\theta)$$ which has vertical asymptotes where $$\theta = k\pi$$ for integers $$k$$, and it decreases from $$+\infty$$ to $$-\infty$$ between asymptotes. 3. **Analyze transformations:** - The argument of cotangent is $$\frac{\pi}{2} x + \frac{3\pi}{4}$$ which means a horizontal compression by a factor of $$\frac{2}{\pi}$$ and a horizontal shift left by $$\frac{3\pi}{4} \times \frac{2}{\pi} = \frac{3}{2}$$ units. - The coefficient $$2$$ in front stretches the graph vertically by a factor of $$2$$. - The $$-3$$ shifts the entire graph down by $$3$$ units. 4. **Shape summary:** The graph retains the cotangent shape: vertical asymptotes, decreasing branches, but shifted and stretched. **Final answer:** The graph is a vertically stretched and shifted cotangent function with vertical asymptotes and decreasing branches, positioned shifted left and down, with the general shape of cotangent.