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

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), so (a = 3), (b = 2)) with a vertical major axis (because (a>b) and the (y)-term has the larger denominator).

Step2: Find the vertices and co - vertices

• For the (y) - axis (major axis) vertices: When (x = 0), we solve for (y): (\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): (\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 vertices ((0,3)), ((0,-3)) and co - vertices ((2,0)), ((-2,0)) on the coordinate plane. Then draw a smooth ellipse passing through these points. The ellipse will be centered at the origin ((0,0)) (since the equation is of the form (\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1) with no shifts in (x) or (y) from the origin).

Answer:

To graph (\boldsymbol{\frac{x^{2}}{4}+\frac{y^{2}}{9}=1}), plot the vertices ((0,3)), ((0, - 3)) and co - vertices ((2,0)), ((-2,0)) and draw a smooth ellipse through these points centered at the origin.