Question

graph each equation.

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

Explanation:

Step1: Identify the form of the equation

The equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) is in the standard form of an ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\) (\(a>b>0\)). Here \(a^{2}=9\), so \(a = 3\), and \(b^{2}=4\), so \(b=2\).

Step2: Find the vertices

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

Step3: Plot the points and draw the ellipse

Plot the vertices \((0,3)\), \((0, - 3)\), \((2,0)\) and \((-2,0)\) on the coordinate plane. Then sketch the ellipse passing through these four points.

Answer:

Plot the points \((0,3)\), \((0, - 3)\), \((2,0)\), \((-2,0)\) and draw an ellipse passing through them.