**Educational Content on Hyperbolas** --- **Question:** Which of the following graphs represents a hyperbola with the equation: \[ \frac{(x - 3)^2}{9} - \frac{(y - 1)^2}{4} = 1 \] **Graph Analysis:** 1. **Top Graph:** - The graph contains two curved lines that open vertically. The curves reflect a hyperbolic shape that opens upwards and downwards. This is typical for hyperbolas centered at a point, with the vertical opening dictated by the equation's structure. - The center of the hyperbola is at the point (3, 1) on the grid. - Asymptotes, while not shown, would typically form a 'cross' through the center, guiding the direction of the curves. 2. **Bottom Graph:** - The graph shows two curves opening horizontally, showcasing a shape characteristic of hyperbolas centered at a specific point, stretching along the horizontal axis. - This does not align with the given equation’s structure as it suggests vertical opening due to \(\frac{(y - 1)^2}{4}\) being subtracted. **Conclusion:** The **top graph** correctly represents the hyperbola as defined by the equation \(\frac{(x - 3)^2}{9} - \frac{(y - 1)^2}{4} = 1\). This hyperbola opens vertically, consistent with the structure of the given equation. ## Graph Analysis for Educational Website ### Graph 1 Description: The top graph depicts a parabolic curve resembling an "n" shape. The parabola opens downwards, with its vertex located approximately at (-0.5, -2). The curve crosses the x-axis around (0, 0) and (-2, 0). The y-axis range is from -10 to 5, while the x-axis extends from -10 to 9. There's a point marked at approximately (-3.5, -1). ### Graph 2 Description: The bottom graph displays a hyperbolic curve. This curve starts in the top-right quadrant and opens towards the top-left, mirrored through the origin into the bottom-left quadrant. It suggests an asymptotic approach towards both axes. Again, the axes range between -10 to 9. A point is marked at approximately (-3.5, -1), consistent with the first graph. ### Key Observations: - **Top Graph (Parabola):** The shape indicates a quadratic function with a downward opening. This implies that the vertex is the maximum point on the curve. - **Bottom Graph (Hyperbola):** The shape indicates the graph of a hyperbolic function, which typically has two distinct branches and asymptotes approaching each axis. These graphs can be explored for their properties, such as vertex, axis of symmetry, roots (for the parabola), and asymptotic behavior (for the hyperbola). For a classroom setting, these graphs are useful for visualizing the behavior of quadratic and hyperbolic functions.