Question

graph each equation.

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

Explanation:

Identify the conic section

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

Compare denominators to find the values of \(a\) and \(b\):

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

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

Since \(b > a\), this is a vertical ellipse.

Determine key points

The center of the ellipse is at \((0, 0)\).
The vertices along the major axis (vertical) are:

$$ (0, \pm b) = (0, \pm 3) $$

The co-vertices along the minor axis (horizontal) are:

$$ (\pm a, 0) = (\pm 2, 0) $$

Sketch the curve

Plot the four key boundary points on the coordinate plane:

• \((2, 0)\)
• \((-2, 0)\)
• \((0, 3)\)
• \((0, -3)\)

Connect these points with a smooth, symmetric oval shape.

Answer:

The equation represents a vertical ellipse centered at the origin \((0,0)\).

• Vertices (y-intercepts): \((0, 3)\) and \((0, -3)\)
• Co-vertices (x-intercepts): \((2, 0)\) and \((-2, 0)\)

To graph the equation, plot these four points on the grid and connect them with a smooth, symmetric curve to form the ellipse.