- Identify the domain \(x \ge -1\) and x-axis symmetry.
- Locate the intercepts at \((-1, 0)\) and \((0, 0)\).
- Characterize the local behavior: the origin \((0,0)\) is a node, and \((-1,0)\) acts as a cusp-like end for a closed loop between \(x=-1\) and \(x=0\).
- Describe the global behavior: two branches for \(x > 0\) that grow as \(\pm x^{3/2}\) for large \(x\).
- Final equation: \(\boxed{y^2 = x^2(x+1)}\).
Explanation
- Domain, Symmetry, and Intercepts
To graph the equation \(y^2 = x^2(x+1)\), we first determine its domain and key characteristics. For \(y^2\) to be non-negative, we require \(x^2(x+1) \ge 0\). Since \(x^2\) is always non-negative, this inequality holds when \(x+1 \ge 0\), meaning the domain is \(x \ge -1\). The equation is invariant when \(y\) is replaced by \(-y\), so the graph is symmetric with respect to the x-axis. Intercepts are found by setting \(y=0\), yielding \(x=0\) and \(x=-1\), resulting in the points \((0, 0)\) and \((-1, 0)\).
- Local Behavior near \((0,0)\) and \((-1,0)\)
Near the origin \((0,0)\), for small \(x\), the equation becomes \(y^2 \approx x^2\), which implies \(y \approx \pm x\). This indicates that the origin is a node where the curve crosses itself. Near \((-1, 0)\), let \(x = -1 + h\) where \(h \ge 0\). Then \(y^2 = (-1+h)^2 h \approx 1^2 \cdot h = h\), so \(y \approx \pm \sqrt{h} = \pm \sqrt{x+1}\). This square root behavior indicates the curve has vertical tangents at \((-1,0)\), creating a cusp-like closure for the loop.
- Global Behavior and Sketching
For \(x > 0\), we have \(y = \pm x\sqrt{x+1}\). As \(x\) increases, \(y\) grows without bound as \(\pm x^{3/2}\). Combining these features, the graph consists of a loop bounded by \(x=-1\) and \(x=0\), and two branches extending from the origin into the first and fourth quadrants that diverge as \(x \to \infty\). The loop exists between \(x=-1\) and \(x=0\), meeting at \((0,0)\) and \((-1,0)\).
- Conclusion and Final Description
The graph of \(y^2 = x^2(x+1)\) is known as the strophoid. It features a closed loop for \(-1 \le x \le 0\) and two unbounded branches for \(x > 0\). The curve passes through \((-1, 0)\) and \((0, 0)\), with the origin being a node where the branches cross, and the point \((-1,0)\) terminating the loop with vertical tangency. The graph is symmetric about the x-axis. \(\boxed{y^2 = x^2(x+1)}\) is the given equation.
Examples
Graphs of complex curves like the strophoid are essential in mechanical design, specifically for mechanisms like the "strophoid" linkage, which was used in antique drawing tools to construct various shapes. Understanding these curves helps engineers design linkages and path-tracing devices, where the geometric properties of the curve—such as its node or loops—determine the constraints and range of motion for robotic arms or specialized machinery.