Question

graph each equation.

1. \frac{x^2}{4} + \frac{y^2}{9} = 1

Explanation:

Identify the conic section type

The given equation is:

$$ \frac{x^2}{4} + \frac{y^2}{9} = 1 $$

This matches the standard form of an ellipse centered at the origin:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

Find the semi-axes lengths

Compare denominators to find \(a\) and \(b\):

$$ a^2 = 4 \implies a = 2 $$

$$ b^2 = 9 \implies b = 3 $$

Since \(b > a\), the major axis is vertical.

Determine the key vertices

The center is at \((0,0)\).
The horizontal vertices (co-vertices) are:

$$ (a, 0) = (2, 0) \quad \text{and} \quad (-a, 0) = (-2, 0) $$

The vertical vertices (vertices) are:

$$ (0, b) = (0, 3) \quad \text{and} \quad (0, -b) = (0, -3) $$

Sketch the ellipse curve

Plot the four key boundary points:

$$ (2,0), (-2,0), (0,3), (0,-3) $$

Connect these points with a smooth, symmetric oval curve.

Answer:

To graph the equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), plot the key points of the ellipse:

• Center: \((0, 0)\)
• Vertices on the y-axis: \((0, 3)\) and \((0, -3)\)
• Co-vertices on the x-axis: \((2, 0)\) and \((-2, 0)\)

Draw a smooth, vertical ellipse passing through these four points.