Question

graph each equation.

1. \\(\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1\\)

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 \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) (where \(a>b>0\) and the major axis is along the \(y\)-axis).

Step2: Find the values of \(a\) and \(b\)

For the given equation, \(a^{2}=9\) so \(a = 3\), and \(b^{2}=4\) so \(b = 2\).

Step3: Determine the vertices and co - vertices

• The vertices of the ellipse (endpoints of the major axis) are at \((0,\pm a)=(0,\pm3)\).
• The co - vertices of the ellipse (endpoints of the minor axis) are at \((\pm b,0)=(\pm2,0)\).

Step4: Plot the points and draw the ellipse

• Plot the points \((0, 3)\), \((0,- 3)\), \((2,0)\) and \((-2,0)\) on the coordinate plane.
• Then, draw a smooth curve connecting these points to form the ellipse. The ellipse will be centered at the origin \((0,0)\) since there are no shifts in the \(x\) or \(y\) terms in the equation.

Answer:

The graph is an ellipse centered at the origin with vertices at \((0, 3)\), \((0,-3)\) and co - vertices at \((2,0)\), \((-2,0)\) (the ellipse is drawn by connecting these points smoothly).