Question

graph each equation.

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

Explanation:

Step1: Identify the type of conic section

The equation \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) is in the form of \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1\) (\(a>b>0\)), which represents an ellipse centered at the origin \((0,0)\).

Step2: Find the vertices

For the ellipse \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), we have \(a^{2}=9\) so \(a = 3\), and \(b^{2}=4\) so \(b=2\).
The vertices along the \(y -\)axis are \((0,a)\) and \((0, - a)\), i.e., \((0,3)\) and \((0,-3)\).
The co - vertices along the \(x -\)axis are \((b,0)\) and \((-b,0)\), i.e., \((2,0)\) and \((-2,0)\).

Step3: Plot the points and draw the ellipse

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

Answer:

Plot the vertices \((0,3)\), \((0,-3)\) and co - vertices \((2,0)\), \((-2,0)\) and draw the ellipse centered at the origin.