graph each equation.\n9) \\(\\frac{x^2}{4} + \\frac{y^2}{9} = 1\\)
graph each equation.\n9) \\(\\frac{x^2}{4} + \\frac{y^2}{9} = 1\\)
Answer
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 and co - vertices
For an ellipse of the form (\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)).
- Vertices: When (x = 0), (\frac{y^{2}}{9}=1), so (y=\pm3). The vertices are ((0,3)) and ((0, - 3)).
- Co - vertices: When (y = 0), (\frac{x^{2}}{4}=1), so (x=\pm2). The co - vertices are ((2,0)) and ((-2,0)).
Step3: Sketch the ellipse
Plot the vertices ((0,3)), ((0, - 3)), ((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 graph is an ellipse with vertices ((0,3)) and ((0, - 3)) and co - vertices ((2,0)) and ((-2,0)).