<< Chapter < Page Chapter >> Page >

Graph the hyperbola given by the equation x 2 144 y 2 81 = 1. Identify and label the vertices, co-vertices, foci, and asymptotes.

vertices: ( ± 12 , 0 ) ; co-vertices: ( 0 , ± 9 ) ; foci: ( ± 15 , 0 ) ; asymptotes: y = ± 3 4 x ;

Got questions? Get instant answers now!

Graphing hyperbolas not centered at the origin

Graphing hyperbolas centered at a point ( h , k ) other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms ( x h ) 2 a 2 ( y k ) 2 b 2 = 1 for horizontal hyperbolas, and ( y k ) 2 a 2 ( x h ) 2 b 2 = 1 for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

Given a general form for a hyperbola centered at ( h , k ) , sketch the graph.

  1. Convert the general form to that standard form. Determine which of the standard forms applies to the given equation.
  2. Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the center, vertices, co-vertices, foci; and equations for the asymptotes.
    1. If the equation is in the form ( x h ) 2 a 2 ( y k ) 2 b 2 = 1 , then
      • the transverse axis is parallel to the x -axis
      • the center is ( h , k )
      • the coordinates of the vertices are ( h ± a , k )
      • the coordinates of the co-vertices are ( h , k ± b )
      • the coordinates of the foci are ( h ± c , k )
      • the equations of the asymptotes are y = ± b a ( x h ) + k
    2. If the equation is in the form ( y k ) 2 a 2 ( x h ) 2 b 2 = 1 , then
      • the transverse axis is parallel to the y -axis
      • the center is ( h , k )
      • the coordinates of the vertices are ( h , k ± a )
      • the coordinates of the co-vertices are ( h ± b , k )
      • the coordinates of the foci are ( h , k ± c )
      • the equations of the asymptotes are y = ± a b ( x h ) + k
  3. Solve for the coordinates of the foci using the equation c = ± a 2 + b 2 .
  4. Plot the center, vertices, co-vertices, foci, and asymptotes in the coordinate plane and draw a smooth curve to form the hyperbola.

Graphing a hyperbola centered at ( h , k ) given an equation in general form

Graph the hyperbola    given by the equation 9 x 2 4 y 2 36 x 40 y 388 = 0. Identify and label the center, vertices, co-vertices, foci, and asymptotes.

Start by expressing the equation in standard form. Group terms that contain the same variable, and move the constant to the opposite side of the equation.

( 9 x 2 36 x ) ( 4 y 2 + 40 y ) = 388

Factor the leading coefficient of each expression.

9 ( x 2 4 x ) 4 ( y 2 + 10 y ) = 388

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

9 ( x 2 4 x + 4 ) 4 ( y 2 + 10 y + 25 ) = 388 + 36 100

Rewrite as perfect squares.

9 ( x 2 ) 2 4 ( y + 5 ) 2 = 324

Divide both sides by the constant term to place the equation in standard form.

( x 2 ) 2 36 ( y + 5 ) 2 81 = 1

The standard form that applies to the given equation is ( x h ) 2 a 2 ( y k ) 2 b 2 = 1 , where a 2 = 36 and b 2 = 81 , or a = 6 and b = 9. Thus, the transverse axis is parallel to the x -axis. It follows that:

  • the center of the ellipse is ( h , k ) = ( 2 , −5 )
  • the coordinates of the vertices are ( h ± a , k ) = ( 2 ± 6 , −5 ) , or ( 4 , −5 ) and ( 8 , −5 )
  • the coordinates of the co-vertices are ( h , k ± b ) = ( 2 , 5 ± 9 ) , or ( 2 , 14 ) and ( 2 , 4 )
  • the coordinates of the foci are ( h ± c , k ) , where c = ± a 2 + b 2 . Solving for c , we have

c = ± 36 + 81 = ± 117 = ± 3 13

Therefore, the coordinates of the foci are ( 2 3 13 , −5 ) and ( 2 + 3 13 , −5 ) .

The equations of the asymptotes are y = ± b a ( x h ) + k = ± 3 2 ( x 2 ) 5.

Next, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the hyperbola, as shown in [link] .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask