Which of the following could be the graph of 4 x squared minus 9 y squared minus 36 equals 0?

Question

(1 point)
Responses

An image shows a circle.
Image with alt text: An image shows a circle.

An image shows a vertical oval.
Image with alt text: An image shows a vertical oval.

An image shows a U-shaped curve opening upward.
Image with alt text: An image shows a U-shaped curve opening upward.

An image shows two curves beside each other opening outward
Image with alt text: An image shows two curves beside each other opening outward
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Answer 1

The equation \(4x^2 - 9y^2 - 36 = 0\) can be rearranged into a standard form.

First, let's isolate the terms involving \(x\) and \(y\):

\[ 4x^2 - 9y^2 = 36 \]

Next, divide everything by 36 to normalize the equation:

\[ \frac{4x^2}{36} - \frac{9y^2}{36} = 1 \]

Simplifying gives:

\[ \frac{x^2}{9} - \frac{y^2}{4} = 1 \]

This is the standard form of a hyperbola:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

where \(a^2 = 9\) (so \(a = 3\)) and \(b^2 = 4\) (so \(b = 2\)).

The graph of this equation will be a hyperbola that opens horizontally.

Given this information, the best choice would be:

• An image shows two curves beside each other opening outward (which represents the hyperbola).

Thus, the correct answer is the image showing two curves beside each other opening outward.