To identify the graph of the given equation, let's analyze the equation step by step:
\[ \frac{(x - 3)^2}{4} - \frac{(y - 6)^2}{9} = 1 \]
This is the standard form of a hyperbola that opens left and right, which typically is represented as:
\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \]
Where:
- \((h, k)\) is the center of the hyperbola,
- \(a^2\) is the denominator for the x-term indicating the distance from the center to the vertices along the x-axis (horizontal),
- \(b^2\) is the denominator for the y-term indicating the distance to the asymptotes.
From your equation:
- \(h = 3\)
- \(k = 6\)
- \(a^2 = 4\) (thus, \(a = 2\)), which gives vertices at \(x = h \pm a = 3 \pm 2\), or \(x = 1\) and \(x = 5\).
- \(b^2 = 9\) (thus, \(b = 3\)), which relates to the slopes of the asymptotes but does not affect the distance of the vertices.
Now that we identify the center and vertices:
- Center: \((3, 6)\)
- Vertices: \((1, 6)\) and \((5, 6)\)
Looking at the provided options, the correct graph is:
A hyperbola opening left and right with center at \((3, 6)\) and vertices \((1, 6)\) and \((5, 6)\).
1 answer