Asked by Jaz
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Question
Which is the graph of the following equation?
Start Fraction left-parenthesis x minus 3 right-parenthesis squared over 4 End Fraction minus Start Fraction left-parenthesis x minus 6 right-parenthesis squared over 9 End Fraction equals 1
(1 point)
Responses
A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 1 comma 6 right-parenthesis and left-parenthesis 5 comma 6 right-parenthesis.
Image with alt text: A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 1 comma 6 right-parenthesis and left-parenthesis 5 comma 6 right-parenthesis.
A hyperbola opening up and down is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 3 comma 3 right-parenthesis and left-parenthesis 3 comma 9 right-parenthesis.
Image with alt text: A hyperbola opening up and down is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 3 comma 3 right-parenthesis and left-parenthesis 3 comma 9 right-parenthesis.
A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 0 comma 6 right-parenthesis and left-parenthesis 6 comma 6 right-parenthesis.
Image with alt text: A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 0 comma 6 right-parenthesis and left-parenthesis 6 comma 6 right-parenthesis.
A hyperbola opening up and down is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 3 comma 4 right-parenthesis and left-parenthesis 3 comma 8 right-parenthesis.
Question
Which is the graph of the following equation?
Start Fraction left-parenthesis x minus 3 right-parenthesis squared over 4 End Fraction minus Start Fraction left-parenthesis x minus 6 right-parenthesis squared over 9 End Fraction equals 1
(1 point)
Responses
A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 1 comma 6 right-parenthesis and left-parenthesis 5 comma 6 right-parenthesis.
Image with alt text: A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 1 comma 6 right-parenthesis and left-parenthesis 5 comma 6 right-parenthesis.
A hyperbola opening up and down is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 3 comma 3 right-parenthesis and left-parenthesis 3 comma 9 right-parenthesis.
Image with alt text: A hyperbola opening up and down is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 3 comma 3 right-parenthesis and left-parenthesis 3 comma 9 right-parenthesis.
A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 0 comma 6 right-parenthesis and left-parenthesis 6 comma 6 right-parenthesis.
Image with alt text: A hyperbola opening left and right is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 0 comma 6 right-parenthesis and left-parenthesis 6 comma 6 right-parenthesis.
A hyperbola opening up and down is graphed on a coordinate grid with center at left-parenthesis 3 comma 6 right-parenthesis and vertices left-parenthesis 3 comma 4 right-parenthesis and left-parenthesis 3 comma 8 right-parenthesis.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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 \((5, 6)\)**.
\[
\frac{(x - 3)^2}{4} - \frac{(y - 6)^2}{9} = 1
\]
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 \((5, 6)\)**.
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