Problem 68 - College Algebra

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 two graphs are not the same because the equation with the absolute values produces four distinct lines opening towards the positive axes, whereas the hyperbola opens to the sides.

Step 1: Graph the Hyperbola

A hyperbola is defined by the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\). In this exercise, the given hyperbola is \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\), where \(a = 4\) and \(b = 3\). To graph this, mark points at \(4, 0\), \(-4, 0\), \(0, 3\) and \(0, -3\), then draw curves tangent to the box defined by these points.

Step 2: Graph the Second Function

Graph the equation \(\frac{x|x|}{16}-\frac{y|y|}{9}=1\). This equation has an absolute value on both \(x\) and \(y\), which means that it will produce 4 different lines, one in each quadrant. To draw them, first separate the equation for each case scenario: (\(x >= 0, y >= 0\); \(x >= 0, y < 0\); \(x < 0, y < 0\); \(x < 0, y >= 0\)). After finding the equation for each quadrant, plot the lines.

Step 3: Compare Both Graphs

The graphs of these two equations look similar but are not exactly the same. The hyperbola opens to the sides, while the second graph produces 4 symmetrical lines in each quadrant, producing a shape that looks like a square rotated 45 degrees. The difference comes from the absolute value in the second equation, which allows both positive and negative outputs, making it always open towards the positive axes.