Question: Graph the following function: $y = -2\csc\left(\frac{4}{7}x + 5\pi\right)$\n\nStep 2 of 2 : Determine how the general shape of the graph, chosen in the previous step, would be shifted, stretched, and reflected for the given function. Graph the results on the axes provided.\n\nAnswer\n\nx- Axis Reflection\n\nReflect graph across x-axis\n\nShift Graph Vertically\n\nUp\nDown\nNone\n\nShift Graph Horizontally (Phase Shift)\n\nLeft\nRight\nNone\n\nStretch/Compress Graph Vertically\n\nYes\nNo\n\nStretch/Compress Graph Horizontally (Period)\n\nYes\nNo\n\nGraph shape: csc curves with upward U-shapes above the x-axis and downward U-shapes below the x-axis, repeating periodically; position_hint=top-right
1. **State the problem:** We need to analyze and graph the function $$y = -2\csc\left(\frac{4}{7}x + 5\pi\right)$$ and determine how it is reflected, shifted, and stretched compared to the basic $$y = \csc x$$ graph.\n\n2. **Recall the general form and properties:** The cosecant function $$y = \csc x$$ has vertical asymptotes where $$\sin x = 0$$, and its graph consists of repeating upward and downward U-shaped curves.\n\n3. **Identify transformations:**\n- The coefficient $$-2$$ reflects the graph across the x-axis and vertically stretches it by a factor of 2.\n- The argument $$\frac{4}{7}x + 5\pi$$ affects horizontal stretch/compression and phase shift.\n\n4. **Horizontal stretch/compression (period):**\nThe period of $$\csc(Bx)$$ is $$\frac{2\pi}{|B|}$$. Here, $$B = \frac{4}{7}$$, so\n$$\text{Period} = \frac{2\pi}{\frac{4}{7}} = 2\pi \times \frac{7}{4} = \frac{14\pi}{4} = \frac{7\pi}{2}.$$\nThis means the graph is horizontally stretched compared to the basic period $$2\pi$$.\n\n5. **Phase shift:**\nThe phase shift is given by solving $$\frac{4}{7}x + 5\pi = 0$$ for $$x$$:\n$$x = -\frac{5\pi}{\frac{4}{7}} = -5\pi \times \frac{7}{4} = -\frac{35\pi}{4}.$$\nThis means the graph shifts to the left by $$\frac{35\pi}{4}$$ units.\n\n6. **Vertical shift:**\nThere is no constant added outside the function, so no vertical shift.\n\n7. **Summary of transformations:**\n- Reflected across the x-axis (due to negative sign).\n- Vertically stretched by factor 2.\n- Horizontally stretched (period changed to $$\frac{7\pi}{2}$$).\n- Shifted left by $$\frac{35\pi}{4}$$.\n- No vertical shift.\n\n8. **Answer to the specific questions:**\n- x-Axis Reflection: Reflect graph across x-axis: **Yes**\n- Shift Graph Vertically: **None**\n- Shift Graph Horizontally (Phase Shift): **Left**\n- Stretch/Compress Graph Vertically: **Yes**\n- Stretch/Compress Graph Horizontally (Period): **Yes**\n\n9. **Graph shape:** The graph consists of periodic $$\csc$$ curves with upward U-shapes above the x-axis and downward U-shapes below the x-axis, reflected and stretched as described, repeating every $$\frac{7\pi}{2}$$ units, shifted left by $$\frac{35\pi}{4}$$.