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 ellipse type
The equation is (\frac{x^{2}}{4}+\frac{y^{2}}{9} = 1), which is in the standard form of an ellipse (\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1) (since (a^{2}=9) and (b^{2}=4), so (a = 3), (b = 2)). The major axis is vertical (because (a^{2}) is under (y^{2})).
Step2: Find the vertices and co - vertices
- For the (y) - axis (major axis) vertices: When (x = 0), we solve for (y) from (\frac{0^{2}}{4}+\frac{y^{2}}{9}=1), so (y^{2}=9), (y=\pm3). So the vertices are ((0, 3)) and ((0, - 3)).
- For the (x) - axis (minor axis) co - vertices: When (y = 0), we solve for (x) from (\frac{x^{2}}{4}+\frac{0^{2}}{9}=1), so (x^{2}=4), (x=\pm2). So the co - vertices are ((2, 0)) and ((- 2, 0)).
Step3: Plot the points and draw the ellipse
Plot the points ((0,3)), ((0, - 3)), ((2,0)) and ((-2,0)) on the coordinate plane. Then, sketch the ellipse passing through these points, with the major axis along the (y) - axis and minor axis along the (x) - axis.
Answer:
The graph is an ellipse centered at the origin ((0,0)) with vertices at ((0, 3)), ((0, - 3)) and co - vertices at ((2, 0)), ((-2, 0)). To draw it, plot these four points and sketch the ellipse connecting them, with the major axis vertical (length (2a = 6)) and minor axis horizontal (length (2b=4)).