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Worksheet: Graphing Hyperbolas

Q1:

The graph shows a sketch of the hyperbola given by the equation 4 𝑦 βˆ’ π‘₯ + 8 𝑦 βˆ’ 1 0 π‘₯ = 2 5 2 2 .

Give the coordinates of the center 𝐢 .

  • A 𝐢 ( βˆ’ 1 , βˆ’ 5 )
  • B 𝐢 ( 5 , 1 )
  • C 𝐢 ( 1 , 5 )
  • D 𝐢 ( βˆ’ 5 , βˆ’ 1 )

Give the coordinates of the vertices 𝑉 1 and 𝑉 2 .

  • A 𝑉 ( βˆ’ 3 , βˆ’ 1 ) 1 , 𝑉 ( βˆ’ 7 , βˆ’ 1 ) 2
  • B 𝑉 ( βˆ’ 4 , βˆ’ 1 ) 1 , 𝑉 ( βˆ’ 6 , βˆ’ 1 ) 2
  • C 𝑉 ( βˆ’ 5 , 1 ) 1 , 𝑉 ( βˆ’ 5 , βˆ’ 3 ) 2
  • D 𝑉 ( βˆ’ 5 , 0 ) 1 , 𝑉 ( βˆ’ 5 , βˆ’ 2 ) 2

Give the coordinates of the foci 𝐹 1 and 𝐹 2 .

  • A 𝐹 ο€» βˆ’ 5 , βˆ’ 1 + √ 5  1 , 𝐹 ο€» βˆ’ 5 , βˆ’ 1 βˆ’ √ 5  2
  • B 𝐹 ο€» βˆ’ 5 + √ 5 , βˆ’ 1  1 , 𝐹 ο€» βˆ’ 5 βˆ’ √ 5 , βˆ’ 1  2
  • C 𝐹 ο€» 5 , βˆ’ 1 + √ 5  1 , 𝐹 ο€» 5 , βˆ’ 1 βˆ’ √ 5  2
  • D 𝐹 ο€» 5 + √ 5 , βˆ’ 1  1 , 𝐹 ο€» 5 βˆ’ √ 5 , βˆ’ 1  2

Give the equations of the asymptotes 𝐴 1 and 𝐴 2 .

  • A 𝐴 𝑦 + 1 = 1 2 ( π‘₯ + 5 ) 1 : , 𝐴 𝑦 + 1 = βˆ’ 1 2 ( π‘₯ + 5 ) 2 :
  • B 𝐴 𝑦 + 1 = 2 ( π‘₯ + 5 ) 1 : , 𝐴 𝑦 + 1 = βˆ’ 2 ( π‘₯ + 5 ) 2 :
  • C 𝐴 𝑦 βˆ’ 1 = 1 2 ( π‘₯ + 5 ) 1 : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 1 2 ( π‘₯ + 5 ) 2 :
  • D 𝐴 𝑦 + 1 = 1 2 ( π‘₯ βˆ’ 5 ) 1 : , 𝐴 𝑦 + 1 = βˆ’ 1 2 ( π‘₯ βˆ’ 5 ) 2 :

Q2:

The graph shows a sketch of the hyperbola given by the equation ( 𝑦 βˆ’ 2 ) 2 5 βˆ’ ( π‘₯ + 3 ) 2 = 1 2 2 .

Give the coordinates of the center 𝐢 .

  • A 𝐢 ( 3 , βˆ’ 2 )
  • B 𝐢 ( 2 , βˆ’ 3 )
  • C 𝐢 ( βˆ’ 2 , 3 )
  • D 𝐢 ( βˆ’ 3 , 2 )

Give the coordinates of the vertices 𝑉 1 and 𝑉 2 .

  • A 𝑉 ο€» βˆ’ 3 + √ 2 , 2  1 , 𝑉 ο€» βˆ’ 3 βˆ’ √ 2 , 2  2
  • B 𝑉 ( 2 , 2 ) 1 , 𝑉 ( βˆ’ 8 , 2 ) 2
  • C 𝑉 ο€» βˆ’ 3 , 2 + √ 2  1 , 𝑉 ο€» βˆ’ 3 , 2 βˆ’ √ 2  2
  • D 𝑉 ( βˆ’ 3 , 7 ) 1 , 𝑉 ( βˆ’ 3 , βˆ’ 3 ) 2

Give the coordinates of the foci 𝐹 1 and 𝐹 2 .

  • A 𝐹 ο€» βˆ’ 3 , 2 + 3 √ 3  1 , 𝐹 ο€» βˆ’ 3 , 2 βˆ’ 3 √ 3  2
  • B 𝐹 ο€» βˆ’ 3 + 3 √ 3 , 2  1 , 𝐹 ο€» βˆ’ 3 βˆ’ 3 √ 3 , 2  2
  • C 𝐹 ο€» 3 , 2 + 3 √ 3  1 , 𝐹 ο€» 3 , 2 βˆ’ 3 √ 3  2
  • D 𝐹 ο€» 3 + 3 √ 3 , 2  1 , 𝐹 ο€» 3 βˆ’ 3 √ 3 , 2  2

Give the equations of the asymptotes 𝐴 1 and 𝐴 2 .

  • A 𝐴 𝑦 βˆ’ 2 = 5 √ 2 ( π‘₯ + 3 ) 1 : , 𝐴 𝑦 βˆ’ 2 = βˆ’ 5 √ 2 ( π‘₯ + 3 ) 2 :
  • B 𝐴 𝑦 βˆ’ 2 = √ 2 5 ( π‘₯ + 3 ) 1 : , 𝐴 𝑦 βˆ’ 2 = βˆ’ √ 2 5 ( π‘₯ + 3 ) 2 :
  • C 𝐴 𝑦 + 2 = 5 √ 2 ( π‘₯ + 3 ) 1 : , 𝐴 𝑦 + 2 = βˆ’ 5 √ 2 ( π‘₯ + 3 ) 2 :
  • D 𝐴 𝑦 βˆ’ 2 = 5 √ 2 ( π‘₯ βˆ’ 3 ) 1 : , 𝐴 𝑦 βˆ’ 2 = βˆ’ 5 √ 2 ( π‘₯ βˆ’ 3 ) 2 :

Q3:

The graph shows a sketch of the hyperbola given by the equation ( π‘₯ βˆ’ 3 ) 4 βˆ’ ( 𝑦 βˆ’ 1 ) 1 6 = 1   .

Give the coordinates of the center 𝐢 .

  • A 𝐢 ( βˆ’ 3 , βˆ’ 1 )
  • B 𝐢 ( 1 , 3 )
  • C 𝐢 ( βˆ’ 1 , βˆ’ 3 )
  • D 𝐢 ( 3 , 1 )

Give the coordinates of the vertices 𝑉  and 𝑉  .

  • A 𝑉 ( 5 , 3 )  , 𝑉 ( βˆ’ 3 , 3 ) 
  • B 𝑉 ( 3 , 3 )  , 𝑉 ( βˆ’ 1 , 3 ) 
  • C 𝑉 ( 7 , 1 )  , 𝑉 ( βˆ’ 1 , 1 ) 
  • D 𝑉 ( 5 , 1 )  , 𝑉 ( 1 , 1 ) 

Give the coordinates of the foci 𝐹  and 𝐹  .

  • A 𝐹 ο€» 3 + 2 √ 5 , 1   , 𝐹 ο€» 3 βˆ’ 2 √ 5 , 1  
  • B 𝐹 ο€» 1 + 2 √ 5 , 3   , 𝐹 ο€» 1 βˆ’ 2 √ 5 , 3  
  • C 𝐹 ο€» βˆ’ 3 + 2 √ 5 , 1   , 𝐹 ο€» βˆ’ 3 βˆ’ 2 √ 5 , 1  
  • D 𝐹 ο€» βˆ’ 1 + 2 √ 5 , 3   , 𝐹 ο€» βˆ’ 1 βˆ’ 2 √ 5 , 3  

Give the equations of the asymptotes 𝐴  and 𝐴  .

  • A 𝐴 𝑦 βˆ’ 1 = 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 2 ( π‘₯ βˆ’ 3 )  :
  • B 𝐴 𝑦 βˆ’ 1 = 1 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 1 2 ( π‘₯ βˆ’ 3 )  :
  • C 𝐴 𝑦 + 1 = 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 + 1 = βˆ’ 2 ( π‘₯ βˆ’ 3 )  :
  • D 𝐴 𝑦 βˆ’ 1 = 2 ( π‘₯ + 3 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 2 ( π‘₯ + 3 )  :

Q4:

The graph shows a sketch of the hyperbola given by the equation 4 π‘₯ βˆ’ 9 𝑦 βˆ’ 1 6 π‘₯ βˆ’ 1 8 2 𝑦 = 2 9 2 2 .

Give the coordinates of the center 𝐢 .

  • A 𝐢 ( βˆ’ 2 , 1 )
  • B 𝐢 ( βˆ’ 1 , 2 )
  • C 𝐢 ( 1 , βˆ’ 2 )
  • D 𝐢 ( 2 , βˆ’ 1 )

Give the coordinates of the vertices 𝑉 1 and 𝑉 2 .

  • A 𝑉 ( 1 , 2 ) , 𝑉 ( βˆ’ 3 , 2 ) 1 2
  • B 𝑉 ( 2 , 2 ) , 𝑉 ( βˆ’ 4 , 2 ) 1 2
  • C 𝑉 ( 4 , βˆ’ 1 ) , 𝑉 ( 0 , βˆ’ 1 ) 1 2
  • D 𝑉 ( 5 , βˆ’ 1 ) , 𝑉 ( βˆ’ 1 , βˆ’ 1 ) 1 2

Give the coordinates of the foci 𝐹 1 and 𝐹 2 .

  • A 𝐹 ο€» 2 + √ 1 3 , βˆ’ 1  , 𝐹 ο€» 2 βˆ’ √ 1 3 , βˆ’ 1  1 2
  • B 𝐹 ο€» 2 , βˆ’ 1 + √ 1 3  , 𝐹 ο€» 2 , βˆ’ 1 βˆ’ √ 1 3  1 2
  • C 𝐹 ο€» βˆ’ 2 + √ 1 3 , βˆ’ 1  , 𝐹 ο€» βˆ’ 2 βˆ’ √ 1 3 , βˆ’ 1  1 2
  • D 𝐹 ο€» βˆ’ 2 , βˆ’ 1 + √ 1 3  , 𝐹 ο€» βˆ’ 2 , βˆ’ 1 βˆ’ √ 1 3  1 2

Give the equations of the asymptotes 𝐴 1 and 𝐴 2 .

  • A 𝐴 ∢ 𝑦 + 1 = 2 3 ( π‘₯ βˆ’ 2 ) , 𝐴 ∢ 𝑦 + 1 = βˆ’ 2 3 ( π‘₯ βˆ’ 2 ) 1 2
  • B 𝐴 ∢ 𝑦 + 1 = 3 2 ( π‘₯ βˆ’ 2 ) , 𝐴 ∢ 𝑦 + 1 = βˆ’ 3 2 ( π‘₯ βˆ’ 2 ) 1 2
  • C 𝐴 ∢ 𝑦 βˆ’ 1 = 2 3 ( π‘₯ βˆ’ 2 ) , 𝐴 ∢ 𝑦 βˆ’ 1 = βˆ’ 2 3 ( π‘₯ βˆ’ 2 ) 1 2
  • D 𝐴 ∢ 𝑦 + 1 = 2 3 ( π‘₯ + 2 ) , 𝐴 ∢ 𝑦 + 1 = βˆ’ 2 3 ( π‘₯ + 2 ) 1 2