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Hyperbola


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Definition of a hyperbola
a set of all points in a plane such that the absolute value of the difference of the distances from two fixed points such as the foci is constant.
How many focus points are in a hyperbola?
The answer is two. Refer to page four in the online lesson for the shape figures and the equation.
Definition of an Asymptote:
a line that a graph of a given function approaches without touching
Fact 1 of an hyperbola:
a hyperbola is a plane that the absolute value of the differences from two fixed points are constant.
Fact 2 of an hyperbola:
Hyperbolas are two curved lines that are in opposite directions.
Fact 3 of an hyperbola:
Hyperbolas are just like an ellipse because it has two foci, axes, and two vertices. (Refer to lesson five for the pictures of what an hyperbola looks like with the vertex, foci, and axis.
Ellipses: denominator switches
Fact 4 of an hyperbola:
Fact 5:
Ellipses have a + sign in their equations
What does variable A represent?
This is very important! The variable A represents distance from center to vertex. Find the length of transverse axis.
What does the variable B represent?
The variable B represents distance from the center to either side of the rectangle, square, etc, that does not contain a vertex.
Identify the center: h, k
Step 1 to solve these problems:
Step 2 to solve these problems:
Identify the distance of the center to vertex, A
Step 3 to solve these problems:
Identify the distance from the center to a side of the associated rectangle, square, etc, without a vertex B.
Step 4 to solve these problems:
Substitute the values of H, K, A, and B into the appropriate form of the equation.
To Graph these equations:
Plot the center: H, K
Second step:
Plot the vertices (Vertex)
Third Step:
Draw the associated Rectangle, square etc.
Fourth Step:
Graph the asymptotes. (Refer to flash card 3 if you forgot the definition of an asymptote.)
Final Step:
Sketch both branches of the hyperbola. You may also want to click on the Graph a Hyperbola activity to learn more about how to graph these.
How can you find the foci of the hyperbola?
The distance from the center of a hyperbola to either focus is c. You can use the equation C^2= A^2 = B^2 to determine C.
What if the transverse axis is horizontal?
Subtract c from, and add c to, the x-coordinate of the center.
What If the transverse axis is vertical?
Subtract c from, and add c to, the y-coordinate of the center.
This is the end of the lesson.
Remember to take the Lesson Quiz when you have concluded this lesson.
Standard Equation {Focal x axis}
{x-h}^2/a^2 - {y-k}^2/b^2 = 1
Standard Equation {focal y axis}
{y-k}^2/a^2 - {x-h}^2/b^2 = 1
Focal Axis
The line through the foci.
{h±c,k}
Foci {Focal x axis}
{h,k±c}
Foci {focal y axis}
{h±a,k}
Vertices {Focal x axis}
{h,k±a}
Vertices {focal y axis}
a
Semitransverse axis
b
Semiconjugate axis
Pythagorean relation
c^2 = a^2 + b^2
y=±{b/a}{x-h}+k
Asymptotes {Focal x axis}
y=±{a/b}{x-h}+k
Asymptotes {focal y axis}
Eccentricity
c/a
Chord
is a segment whose endpoints lie on the circle
foci
the two fixed points
vertices
The perpendicular bisectors of a triangle intersect at a point that is equidistant from "these" of the triangle
Center
Each regular polygon has a center because it can be inscribed in a circle.
Transverse Axis
The axis that goes through the vertices of a hyperbola.
conjugate axis
the line segment perpendicular to the transverse axis. its length is 2b
latis rectum
the width of each branch at the focus
c
>a
Hyperbola:
The set of points whose distances from two fixed points (foci) differ by a constant. A hyperbola has two halves, and each half has a vertex.
Sides
The angle bisectors of a triangle intersect at a point that is equidistant from all of "these" of the triangle
X-oriented hyperbola standard equation:
(x-h)^2 / a^2 - (y-k)^2 / b^2 =1
(h,k)
Center of a ellipse
(h-a,k) and (h+a,k)
X-Oriented hyperbola vertices Are:
Transverse Axis:
The segment joining the two vertices. The transverse axis is horizontal and has length 2a.
X-oriented hyperbola foci:
Located on the transverse axis. (h-c,k) and (h+c,k)
c=√(a^2 +b^2)
Distance between center and each focus is:
vertical
The direction of something that runs north to south or south to north.
X-oriented hyperbola Asymptote Equations:
y-k = ± b/a (x-h)
Y-oriented hyperbola opens:
Up or down
Y-oriented hyperbola standard equation:
(y-k)^2 / a^2 - (x-h)^2 / b^2 =1
Y-oriented hyperbola vertices:
(h,k-a) and (h,k+a)
Y-oriented hyperbola transverse axis:
Is vertical and has length 2a
Y-oriented hyperbola foci:
(h,k-c) and (h,k+c)
Horizontal
Straight line across from side to side (length)
Y-oriented hyperbola asymptote equation:
y-k = ± a/b (x-h)
the hyperbola has an x-orientation
If the x-term of the equation is positive,
the hyperbola has a y-orientation
If the y-term of the equation is positive,