91Ó°ÊÓ

Conic sections are curves obtained by intersecting a plane with a conic surface. The primary forms of conic sections you will encounter include circles, ellipses, parabolas, and hyperbolas. Each has a unique equation that can be written in a 'standard form'. This helps in easily identifying the type of curve and its key features.

Ellipses can be seen in real life, such as the orbit of planets. Algebraically, an ellipse is a set of points where the sum of the distances to two fixed points (called foci) is constant. The standard form of the ellipse equation is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]. Here, \a\ represents the semi-major axis length, and \b\ represents the semi-minor axis length. This means that the ellipse extends from \(-a\) to \a\ along the x-axis and from \(-b\) to \b\ along the y-axis. For our exercise, we transformed the given inequality \(y^2 + 2x^2 \leq 4\). Into its elliptical form \[ \frac{y^2}{4} + \frac{x^2}{2} \leq 1 \]. This helps identify the ellipse's orientation and the lengths of its axes.

Graphing inequalities involves understanding both the boundary of the region defined by the inequality and which side of the boundary represents the solution set. When dealing with conic sections, such as ellipses, the boundary is the curve itself. From our exercise, once we converted the inequality to its standardized elliptical form \[ \frac{y^2}{4} + \frac{x^2}{2} \leq 1 \], we plotted the ellipse by identifying key points (vertices and co-vertices).
After plotting the ellipse, the final step is shading the appropriate region. Because our inequality sign is \( \leq\)\, we shade inside the ellipse.
This shading indicates all points within this region satisfy the given inequality, thus providing a clear visual solution.