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 \(\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\) (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): The vertices are at \((0,\pm a)=(0,\pm3)\).
• For the \(x\) - axis (minor axis): The co - vertices are at \((\pm b,0)=(\pm2,0)\).

Step3: Plot the points

• Plot the vertices \((0,3)\), \((0, - 3)\) and the co - vertices \((2,0)\), \((- 2,0)\).
• Then, sketch the ellipse by connecting these points smoothly, making sure that the ellipse is symmetric about both the \(x\) - axis and \(y\) - axis.

Answer:

The graph is an ellipse with vertices at \((0,3)\), \((0, - 3)\) and co - vertices at \((2,0)\), \((- 2,0)\), sketched on the given coordinate grid by connecting these points smoothly. (To present the final graph, we can describe it as above. If we were to draw it, we would plot the four points and draw the ellipse passing through them.)