Question

graph each equation.

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

Explanation:

🆕 New Concept Discovered: Graphing Ellipses
An ellipse is a stretched circle with a horizontal and vertical reach.

Step 1: Identify the conic section

The given equation is:

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

This fits the standard form of an ellipse centered at the origin \((0,0)\):

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

Step 2: Find the horizontal and vertical stretches

Identify the denominators to find how far the graph extends from the center:

• Horizontal stretch factor \(a\):

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

This means the graph extends \(2\) units to the left and right of the center.

• Vertical stretch factor \(b\):

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

This means the graph extends \(3\) units up and down from the center.

Step 3: Plot the key points

Using the center \((0,0)\), plot the four vertices:

• Left and right vertices: \((2, 0)\) and \((-2, 0)\)
• Top and bottom vertices: \((0, 3)\) and \((0, -3)\)

Step 4: Draw the curve

Sketch a smooth, oval-shaped curve passing through the four plotted points: \((2,0)\), \((-2,0)\), \((0,3)\), and \((0,-3)\).

Answer:

To graph the equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), plot the following four points on the grid and connect them with a smooth ellipse (oval):

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

The resulting graph is a vertically oriented ellipse centered at \((0,0)\).