How to graph a hyperbola?
Understand the Problem
The question is asking how to graph a hyperbola, which involves understanding its standard equation, identifying its key features such as the vertices and foci, and plotting it accordingly on a coordinate plane.
Answer
To graph a hyperbola, identify the center $(h, k)$, the vertices, calculate the foci with $c = \sqrt{a^2 + b^2}$, draw asymptotes, and sketch accordingly.
Answer for screen readers
To graph a hyperbola, identify the center $(h, k)$, vertices at $(h \pm a, k)$ or $(h, k \pm a)$, foci using $c = \sqrt{a^2 + b^2}$, draw a box, and sketch the hyperbola near its asymptotes.
Steps to Solve
- Identify the standard form of a hyperbola Hyperbolas can be represented in two standard forms:
- Horizontal hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
- Vertical hyperbola: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$
Here, $(h, k)$ is the center of the hyperbola, and $a$ and $b$ are the distances from the center to the vertices and co-vertices respectively.
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Determine the center of the hyperbola From the standard equation, identify the values of $h$ and $k$. The center is located at the point $(h, k)$.
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Find the vertices The vertices are located at a distance of $a$ from the center.
- For horizontal hyperbolas, the vertices will be at $(h \pm a, k)$.
- For vertical hyperbolas, the vertices will be at $(h, k \pm a)$.
- Locate the foci The foci are calculated using the equation $c = \sqrt{a^2 + b^2}$. The foci are located at a distance of $c$ from the center.
- For horizontal hyperbolas, the foci will be at $(h \pm c, k)$.
- For vertical hyperbolas, the foci will be at $(h, k \pm c)$.
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Draw the rectangular box To aid in accurately sketching the hyperbola, draw a rectangle with corners at the vertices, then extend diagonally to locate the asymptotes. The slopes for the asymptotes are $\pm \frac{b}{a}$.
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Sketch the hyperbola Finally, draw the hyperbola that approaches the lines of the asymptotes, ensuring it passes through the vertices and is centered at $(h, k)$.
To graph a hyperbola, identify the center $(h, k)$, vertices at $(h \pm a, k)$ or $(h, k \pm a)$, foci using $c = \sqrt{a^2 + b^2}$, draw a box, and sketch the hyperbola near its asymptotes.
More Information
Hyperbolas represent a type of conic section, and they can be found in real-world situations such as in the paths of certain celestial bodies or in some types of reflective surfaces. They have unique properties that make them interesting in the study of mathematics and physics.
Tips
- Forgetting to calculate $c$ correctly when finding the foci, which results in wrong focus placement.
- Misidentifying whether the hyperbola is horizontal or vertical based on the equation.
- Not plotting the asymptotes correctly, leading to an inaccurate shape of the hyperbola.
