Hyperbolas are fascinating curves with two separate branches that extend infinitely. They're defined by a set of points where the difference in distances from two fixed points (foci ) is constant. This unique property gives hyperbolas their distinctive shape and useful applications.
Understanding hyperbolas involves key concepts like foci , vertices , and asymptotes . These elements help us graph hyperbolas and solve real-world problems in fields like physics, astronomy, and engineering. Mastering hyperbola equations and their relationships to graphs is crucial for analyzing their behavior.
Hyperbola Fundamentals
Key points of hyperbolas
Top images from around the web for Key points of hyperbolas The Hyperbola | Algebra and Trigonometry View original
Is this image relevant?
The Hyperbola · Algebra and Trigonometry View original
Is this image relevant?
The Hyperbola · Precalculus View original
Is this image relevant?
The Hyperbola | Algebra and Trigonometry View original
Is this image relevant?
The Hyperbola · Algebra and Trigonometry View original
Is this image relevant?
1 of 3
Top images from around the web for Key points of hyperbolas The Hyperbola | Algebra and Trigonometry View original
Is this image relevant?
The Hyperbola · Algebra and Trigonometry View original
Is this image relevant?
The Hyperbola · Precalculus View original
Is this image relevant?
The Hyperbola | Algebra and Trigonometry View original
Is this image relevant?
The Hyperbola · Algebra and Trigonometry View original
Is this image relevant?
1 of 3
A hyperbola is defined by two points called foci
Located on the transverse axis , equidistant from the center
Distance from the center to a focus denoted as [ c ] ( h t t p s : / / w w w . f i v e a b l e K e y T e r m : c ) [c](https://www.fiveableKeyTerm:c) [ c ] ( h ttp s : // www . f i v e ab l eKey T er m : c ) (focal length )
Vertices are the points where the hyperbola intersects the transverse axis
Distance from the center to a vertex denoted as a a a
Transverse axis is the line segment that passes through the foci and vertices
Conjugate axis is perpendicular to the transverse axis and passes through the center
Center is the midpoint of the line segment connecting the foci
Intersection point of the transverse and conjugate axes
Latus rectum is a line segment perpendicular to the transverse axis, passing through a focus
Hyperbola Equations and Graphing
Standard form depends on the orientation of the transverse axis
Horizontal transverse axis (opens left and right): ( x − h ) 2 a 2 − ( y − k ) 2 [ b ] ( h t t p s : / / w w w . f i v e a b l e K e y T e r m : b ) 2 = 1 \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{[b](https://www.fiveableKeyTerm:b)^2} = 1 a 2 ( x − h ) 2 − [ b ] ( h ttp s : // www . f i v e ab l eKey T er m : b ) 2 ( y − k ) 2 = 1
Vertical transverse axis (opens up and down): ( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1 \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 a 2 ( y − k ) 2 − b 2 ( x − h ) 2 = 1
( h , k ) (h, k) ( h , k ) represents the center of the hyperbola
Constants a a a and b b b determine the shape
a a a is the distance from the center to a vertex
b b b is related to the distance from the center to the conjugate axis
Relationship between a a a , b b b , and c c c (distance from center to focus) given by c 2 = a 2 + b 2 c^2 = a^2 + b^2 c 2 = a 2 + b 2
Pythagorean theorem connects these key distances
Graphing with various centers
Steps to graph a hyperbola:
Identify and plot the center ( h , k ) (h, k) ( h , k ) on the coordinate plane
Determine the transverse axis orientation (horizontal or vertical)
Calculate and plot the vertices using a a a
Calculate and plot the co-vertices using b b b
Calculate and plot the foci using c c c
Draw the hyperbola passing through the vertices and approaching the asymptotes
Asymptotes are lines that the hyperbola approaches but never touches
Pass through the center with slopes of ± b a \pm \frac{b}{a} ± a b
Act as boundaries for the hyperbola's arms
Applications and Analysis
Real-world applications of hyperbolas
Physics: modeling paths of objects moving faster than the speed of sound (supersonic flight)
Astronomy: describing orbits of comets and other celestial bodies around the sun
Engineering: designing reflective surfaces for satellite dishes and telescopes (parabolic reflectors)
To solve real-world problems:
Identify given information and match it to the hyperbola equation
Determine the center, vertices, and foci based on the problem context
Use the equation to calculate distances, speeds, or other relevant quantities
Relationships between equation and graph
Center ( h , k ) (h, k) ( h , k ) determines the location on the coordinate plane
Shifting the center translates the entire hyperbola
Constants a a a and b b b affect the shape and orientation
As a a a increases, the hyperbola becomes narrower horizontally
As b b b increases, the hyperbola becomes narrower vertically
Foci and vertices are symmetrically located around the center
Distance between foci is 2 c 2c 2 c , distance between vertices is 2 a 2a 2 a
Asymptotes represent the direction in which the hyperbola extends indefinitely
Slopes of asymptotes determined by the ratio b a \frac{b}{a} a b
Larger b a \frac{b}{a} a b results in steeper asymptotes
Hyperbolas are one of the four conic sections , along with circles, ellipses, and parabolas
Each conic section has a directrix , which is a line used in its geometric definition
The shape of a conic section is determined by its eccentricity , which relates to the ratio of distances from any point on the curve to a focus and the directrix