Problem 80 - Algebra and Trigonometry

Question

Graph \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\) and \(\frac{x|x|}{16}-\frac{y|y|}{9}=1\) in the same viewing rectangle. Explain why the graphs are not the same.

Step-by-Step Solution

Answer

The first equation graphs a hyperbola in all four quadrants. The second equation graphs a hyperbola only in the first quadrant due to the absolute value operation, which restricts the equation to nonnegative values of \(x\) and \(y\). Therefore, the graphs of both equations are not the same.

Step 1: Graphing the first equation

Begin by drawing the graph of the equation \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\) which is a hyperbola with the center at the origin. The vertices of this hyperbola would be at \((\pm 4, 0)\) and the foci would be at \((\pm 5, 0)\). Draw this hyperbola.

Step 2: Graphing the second equation

Next, focusing on graphing the second equation \(\frac{x|x|}{16}-\frac{y|y|}{9}=1\). Given that \(x|x|\) is always positive for all \(x\), this means that this equation will be the same as the first equation for \(x > 0\) and \(y > 0\) and will be zero otherwise. Now graph this equation, noting the differences from the first graph.

Step 3: Observations and Explanation

Both graphs should be compared. The graph from the second equation will have only the first quadrant portion of the graph from the first equation because, for negative values of \(x\) and \(y\), the equation reduces to zero. This explains why the graphs of these two equations are not the same.