Question

graph each equation.

1. \\( \frac { x ^ { 2 } } { 4 } + \frac { 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 \(a>b\) and the major axis is along the \(y\)-axis). Here, \(a^{2}=9\) so \(a = 3\), and \(b^{2}=4\) so \(b = 2\).

Step2: Find the vertices and co - vertices

• For the \(y\)-intercepts (vertices, since major axis is along \(y\)-axis), set \(x = 0\). Then \(\frac{y^{2}}{9}=1\), so \(y^{2}=9\) and \(y=\pm3\). So the points are \((0, 3)\) and \((0, - 3)\).
• For the \(x\)-intercepts (co - vertices), set \(y = 0\). Then \(\frac{x^{2}}{4}=1\), so \(x^{2}=4\) and \(x=\pm2\). So the points 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 draw a smooth curve connecting these points to form the ellipse. The ellipse will be centered at the origin \((0,0)\), stretched 3 units up and down along the \(y\)-axis and 2 units left and right along the \(x\)-axis.

Answer:

To graph \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\):

1. Recognize it is an ellipse centered at \((0,0)\) with \(a = 3\) (along \(y\)-axis) and \(b=2\) (along \(x\)-axis).
2. Plot vertices \((0,3)\), \((0, - 3)\) and co - vertices \((2,0)\), \((-2,0)\).
3. Draw a smooth ellipse through these points. (The graph is an ellipse centered at the origin, with major axis along the \(y\)-axis, passing through \((0,\pm3)\) and \((\pm2,0)\))