Question

sketch a graph of \\(\frac{x^2}{4} + \frac{y^2}{36} = 1\\)

Explanation:

Step 1: Identify the conic section type

The equation

$$\frac{x^2}{4} + \frac{y^2}{36} = 1$$

represents an ellipse centered at the origin \((0,0)\).

Step 2: Find the horizontal intercepts

Set \(y = 0\) to find the \(x\)-intercepts:

$$\frac{x^2}{4} = 1 \implies x = \pm 2$$

Step 3: Find the vertical intercepts

Set \(x = 0\) to find the \(y\)-intercepts:

$$\frac{y^2}{36} = 1 \implies y = \pm 6$$

Step 4: Sketch the ellipse

Plot the vertices \((2,0)\), \((-2,0)\), \((0,6)\), and \((0,-6)\) and draw a smooth ellipse through them.

Answer:

An ellipse centered at \((0,0)\) with horizontal vertices at \((-2, 0)\) and \((2, 0)\), and vertical vertices at \((0, -6)\) and \((0, 6)\).