Question

Graph the equation: $x^{2}+4 y^{2}=4$.

   Graph the equation: $x^{2}+4 y^{2}=4$.

Precalculus

Michael Sullivan 9th Edition

Chapter 11, Problem 4

Instant Answer

Solved by Expert Grace Bajar

07/06/2020

Step 1

We can rewrite this equation in the standard form of an ellipse by dividing each term by 4. This gives us $\frac{x^{2}}{4} + y^{2} = 1$. Show more…

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Graph the equation: $x^{2}+4 y^{2}=4$.

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Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (the foci) is constant. It typically appears as a closed, oval-shaped curve with two axes of symmetry and is one of the primary shapes studied in analytic geometry.

Standard Form of an Ellipse Equation

Rewriting the equation in standard form is essential for identifying the key features of an ellipse, such as its center, and the lengths of its semi-major and semi-minor axes. Transforming an equation like x² + 4y² = 4 by dividing through by a constant helps in recognizing the structure of the ellipse in the form (x-h)²/a² + (y-k)²/b² = 1.

Graphing Techniques for Ellipses

Graphing an ellipse involves first expressing the equation in a form that clearly shows the center and the lengths of the axes, then plotting these essential points and using the symmetry of the ellipse to sketch its curve. This process includes determining intercepts, vertices, and co-vertices, which collectively assist in accurately drawing the ellipse on a coordinate plane.

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