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 stretch.

Step 1: Identify the conic section and its center

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 \( (0,0) \):

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

Since there are no numbers subtracted from \( x \) or \( y \) inside the squared terms, the center of this ellipse is at \( (0,0) \).

Step 2: Find the horizontal and vertical intercepts

To find how far the ellipse stretches horizontally and vertically from the center:

• Horizontal stretch (\( x \)-direction):

The denominator under \( x^2 \) is \( 4 \).

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

This means the ellipse extends \( 2 \) units to the left and right of the center.
The \( x \)-intercepts are at \( (2, 0) \) and \( (-2, 0) \).

• Vertical stretch (\( y \)-direction):

The denominator under \( y^2 \) is \( 9 \).

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

This means the ellipse extends \( 3 \) units up and down from the center.
The \( y \)-intercepts are at \( (0, 3) \) and \( (0, -3) \).

Step 3: Plot the key points and draw the curve

1. Plot the center at \( (0,0) \).
2. Plot the horizontal vertices on the \( x \)-axis: \( (2,0) \) and \( (-2,0) \).
3. Plot the vertical vertices on the \( y \)-axis: \( (0,3) \) and \( (0,-3) \).
4. Connect these four points with a smooth, rounded oval shape (ellipse).

Answer:

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

• Right vertex: \( (2, 0) \)
• Left vertex: \( (-2, 0) \)
• Top vertex: \( (0, 3) \)
• Bottom vertex: \( (0, -3) \)