graph each equation.\n9) \\(\\dfrac{x^2}{4} + \\dfrac{y^2}{9} = 1\\)\ncoordinate grid with x from -8 to 8…

Explanation:

Step1: Identify the conic section

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) (since (a^{2}=9) and (b^{2} = 4), (a>b), so it is a vertical ellipse). Here, the center of the ellipse is at the origin ((0,0)) because there are no shifts in (x) or (y) (the numerators are (x^{2}) and (y^{2}) without any linear terms).

Step2: Find the vertices and co - vertices

For a vertical ellipse (\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1), the length of the semi - major axis (a=\sqrt{9}=3) and the length of the semi - minor axis (b = \sqrt{4}=2). - The vertices (endpoints of the major axis) are at ((0,\pm a)=(0,\pm3)). - The co - vertices (endpoints of the minor axis) are at ((\pm b,0)=(\pm2,0)).

Step3: Plot the points

Plot the center ((0,0)), the vertices ((0,3)), ((0, - 3)) and the co - vertices ((2,0)), ((- 2,0)). Then, draw a smooth ellipse passing through these points. The ellipse will be taller along the (y) - axis (since the major axis is along the (y) - axis) with a width of (2b = 4) (from (x=-2) to (x = 2)) and a height of (2a=6) (from (y=-3) to (y = 3)).

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)), and it is drawn by connecting these points with a smooth curve. (To actually draw it, plot the points ((0,3)), ((0, - 3)), ((2,0)), ((-2,0)) on the given coordinate grid and sketch the ellipse through them.)