Math Problem Statement
Solution
The equation given in the image is:
This is the standard form of a hyperbola. The general form of a hyperbola centered at the origin is:
From comparing the two equations, we have:
- , so
- , so
Important Features of the Hyperbola:
- Center: because the equation is in its standard form and there is no translation.
- Vertices: The vertices are located along the x-axis at , which in this case gives the vertices at .
- Co-vertices: The co-vertices are along the y-axis at , which gives . These help in defining the rectangle that guides the asymptotes.
- Asymptotes: The asymptotes of the hyperbola are lines that pass through the center, with slopes given by . The asymptotes are:
Graphing the Hyperbola:
- Start by plotting the center at .
- Plot the vertices at and .
- Plot the co-vertices at and .
- Draw a rectangle using the vertices and co-vertices as guides. The diagonals of this rectangle are the asymptotes of the hyperbola.
- Sketch the hyperbola, which opens horizontally since the positive term is with , and approaches the asymptotes as it moves away from the center.
Would you like any further details or explanations?
Follow-up questions:
- How do we determine if a hyperbola opens horizontally or vertically?
- Can we apply the same method if the hyperbola is not centered at the origin?
- How do asymptotes help in sketching a hyperbola?
- What is the significance of the foci of a hyperbola?
- How would the equation change if the hyperbola opened vertically?
Tip: When graphing conic sections like hyperbolas, identifying the center, vertices, and asymptotes is crucial to sketching an accurate graph.
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Math Problem Analysis
Mathematical Concepts
Conic Sections
Hyperbolas
Coordinate Geometry
Formulas
Standard form of a hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Vertices: \((\pm a, 0)\) or \((0, \pm b)\)
Asymptotes: \(y = \pm \frac{b}{a}x\)
Theorems
Properties of Hyperbolas
Suitable Grade Level
Grades 10-12
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