Question

graph each equation.

1. \\( \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1 \\)

Explanation:

Step1: Identify the standard form of the ellipse

The standard form of an ellipse is \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\) (\(a>b>0\)), where the major axis is along the \(y\) - axis. For the equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\), we have \(b^{2}=4\), so \(b = 2\), and \(a^{2}=9\), so \(a=3\).

Step2: Find the vertices and co - vertices

The vertices of the ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) are \((0,\pm a)\) and the co - vertices are \((\pm b,0)\).
Since \(a = 3\) and \(b = 2\), the vertices are \((0,3)\) and \((0, - 3)\), and the co - vertices are \((2,0)\) and \((-2,0)\).

Step3: Sketch the ellipse

Plot the vertices \((0,3)\), \((0, - 3)\), co - vertices \((2,0)\) and \((-2,0)\) on the coordinate plane. Then draw a smooth curve passing through these four points to form the ellipse.

Answer:

The ellipse has vertices \((0,3)\) and \((0, - 3)\), co - vertices \((2,0)\) and \((-2,0)\). Sketch the ellipse by connecting these four points with a smooth curve.