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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), when \(x = 0\), we have \(\frac{y^{2}}{9}=1\), so \(y=\pm3\). The vertices (end - points of the major axis) are \((0,3)\) and \((0, - 3)\).
When \(y = 0\), we have \(\frac{x^{2}}{4}=1\), so \(x=\pm2\). The co - vertices (end - points of the minor axis) are \((2,0)\) 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 points \((0,3)\), \((0, - 3)\), \((2,0)\), \((-2,0)\) and draw an ellipse passing through them.