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Worksheet: Properties of Parabolas

Q1:

Find the coordinates of the vertex of the function 𝑓 ( π‘₯ ) = βˆ’ 8 π‘₯ βˆ’ 6 + 4 π‘₯ 2 .

  • A ( βˆ’ 1 , βˆ’ 5 )
  • B ( 0 , βˆ’ 6 )
  • C ( 2 , βˆ’ 1 0 )
  • D ( 1 , βˆ’ 1 0 )

Q2:

The figure shows the curve ( 𝑦 + π‘₯ βˆ’ 3 ) + 4 ( 𝑦 βˆ’ π‘₯ ) + 4 = 0 2 , the dashed line 𝑦 + π‘₯ = 3 , and its perpendicular 𝑦 βˆ’ π‘₯ = βˆ’ 5 .

Determine the coordinates of the two intersections 𝐴 and 𝐡 .

  • A 𝐴 = ( 4 , βˆ’ 1 ) , 𝐡 = ( 1 , 2 )
  • B 𝐴 = ( 3 , βˆ’ 2 ) , 𝐡 = ( 1 , 6 )
  • C 𝐴 = ( 1 , βˆ’ 4 ) , 𝐡 = ( 2 , 1 )
  • D 𝐴 = ( 2 , βˆ’ 3 ) , 𝐡 = ( 6 , 1 )
  • E 𝐴 = ( 2 , βˆ’ 3 ) , 𝐡 = ( 4 , 1 )

The vertex of this parabola lies where the line 𝑦 βˆ’ π‘₯ = 𝐾 meets the parabola at exactly one point.

What is the value of 𝐾 ? What are the coordinates of the vertex?

  • A 𝐾 = βˆ’ 1 , vertix: ( 2 , 1 )
  • B 𝐾 = 2 , vertix: ( 1 , 2 )
  • C 𝐾 = βˆ’ 2 , vertix: ( 1 , 2 )
  • D 𝐾 = 1 , vertix: ( 2 , 1 )
  • E 𝐾 = βˆ’ 1 , vertix: ( 3 , 1 )

The vertex also lies on the line of symmetry 𝑦 + π‘₯ = 3 , which is present in the equation. It turns out that the equation ( 𝑦 + π‘₯ βˆ’ 3 ) βˆ’ 8 ( 𝑦 + π‘₯ βˆ’ 3 ) + 4 ( 𝑦 βˆ’ π‘₯ ) + 2 = 0 2 gives a parabola whose axis of symmetry is parallel to 𝑦 + π‘₯ = 3 .

By completing the square and rewriting this as ( 𝑦 + π‘₯ βˆ’ 𝐴 ) + 4 ( 𝑦 βˆ’ π‘₯ ) + 𝐡 = 0 2 , determine what the new axis of symmetry is. What is the value of the constant 𝐡 ?

  • A 𝑦 + π‘₯ = 1 , 𝐡 = 2
  • B 𝑦 + π‘₯ = 4 , 𝐡 = βˆ’ 1 4
  • C 𝑦 + π‘₯ = 7 , 𝐡 = βˆ’ 1 4
  • D 𝑦 + π‘₯ = 1 , 𝐡 = βˆ’ 3
  • E 𝑦 + π‘₯ = 4 , 𝐡 = 2

By determining 𝐴 , 𝐡 such that 𝑦 βˆ’ 5 π‘₯ = 𝐴 ( 𝑦 + π‘₯ ) + 𝐡 ( 𝑦 βˆ’ π‘₯ ) , complete the squares and rewrite ( 𝑦 + π‘₯ ) + 𝑦 βˆ’ 5 π‘₯ + 7 2 as ( 𝑦 + π‘₯ βˆ’ π‘Ž ) + 𝑏 ( 𝑦 βˆ’ π‘₯ ) + 𝑐 2 . What is this expression? What is the vertex of the parabola?

  • A ( 𝑦 + π‘₯ βˆ’ 1 ) + 3 ( 𝑦 βˆ’ π‘₯ ) + 6 2 , ο€Ό 3 2 , βˆ’ 1 2 
  • B ( 𝑦 + π‘₯ βˆ’ 1 ) + 3 ( 𝑦 βˆ’ π‘₯ ) + 7 2 , ο€Ό 3 2 , βˆ’ 1 2 
  • C ( 𝑦 + π‘₯ + 1 ) + 3 ( 𝑦 βˆ’ π‘₯ ) + 6 2 , ο€Ό βˆ’ 3 2 , 1 2 
  • D ( 𝑦 + π‘₯ + 1 ) βˆ’ 3 ( 𝑦 βˆ’ π‘₯ ) + 6 2 , ο€Ό βˆ’ 3 2 , 1 2 
  • E ( 𝑦 + π‘₯ βˆ’ 1 ) βˆ’ 3 ( 𝑦 βˆ’ π‘₯ ) + 6 2 , ο€Ό 3 2 , βˆ’ 1 2 

Q3:

A parabola has an equation ( π‘₯ βˆ’ 4 ) = 2 0 ( 𝑦 + 3 ) 2 .

Find the coordinates of the vertex.

  • A ( 4 , 2 )
  • B ( βˆ’ 4 , 3 )
  • C ( βˆ’ 4 , βˆ’ 2 )
  • D ( 4 , βˆ’ 3 )
  • E ( 8 , βˆ’ 3 )

Determine the equation of the directrix.

  • A 𝑦 = βˆ’ 8
  • B 𝑦 = βˆ’ 1
  • C 𝑦 = 9
  • D 𝑦 = 2
  • E 𝑦 = βˆ’ 2 3

Q4:

Consider the parabola with Cartesian equation 𝑦 = βˆ’ 2 √ 1 7 π‘₯ 2 .

What are the coordinates of its focus?

  • A ο€» βˆ’ 2 √ 1 7 , 0 
  • B ο€Ώ 0 , βˆ’ √ 1 7 2 
  • C ο€» 0 , βˆ’ 2 √ 1 7 
  • D ο€Ώ βˆ’ √ 1 7 2 , 0 
  • E ο€Ώ 0 , √ 1 7 2 

Write the equation of its directrix.

  • A π‘₯ = √ 1 7 2
  • B π‘₯ = βˆ’ 2 √ 1 7
  • C π‘₯ = 2 √ 1 7
  • D π‘₯ = βˆ’ √ 1 7 2
  • E π‘₯ = 8 √ 1 7

Q5:

Find the vertex of the graph of 𝑦 = ( 3 π‘₯ + 1 ) 2 .

  • A ο€Ό 0 , βˆ’ 1 3 
  • B ο€Ό 1 3 , 0 
  • C ο€Ό 0 , 1 3 
  • D ο€Ό βˆ’ 1 3 , 0 
  • E ο€Ό βˆ’ 1 3 , βˆ’ 1 3 

Q6:

Consider the parabola with the Cartesian equation π‘₯ = βˆ’ 6 √ 1 1 𝑦 2 .

What are the coordinates of the focus of the parabola with the Cartesian equation π‘₯ = βˆ’ 6 √ 1 1 𝑦 2 ?

  • A ο€» 0 , βˆ’ 6 √ 1 1 
  • B ο€Ώ βˆ’ 3 √ 1 1 2 , 0 
  • C ο€» βˆ’ 6 √ 1 1 , 0 
  • D ο€Ώ 0 , βˆ’ 3 √ 1 1 2 
  • E ο€» 0 , βˆ’ 2 4 √ 1 1 

Write the equation of its directrix.

  • A 𝑦 = 3 √ 1 1 2
  • B 𝑦 = βˆ’ 6 √ 1 1
  • C 𝑦 = 6 √ 1 1
  • D 𝑦 = βˆ’ 3 √ 1 1 2
  • E 𝑦 = 2 4 √ 1 1

Q7:

Find the vertex of the graph of 𝑦 = ( π‘₯ βˆ’ 1 ) 2 .

  • A ( 0 , 1 )
  • B ( βˆ’ 1 , 0 )
  • C ( 0 , βˆ’ 1 )
  • D ( 1 , 0 )
  • E ( 1 , 1 )

Q8:

Find the point of symmetry of the curve of the function 𝑓 ( π‘₯ ) = βˆ’ 7 βˆ’ ( π‘₯ βˆ’ 2 ) 2 .

  • A ( βˆ’ 2 , βˆ’ 7 )
  • B ( βˆ’ 7 , 2 )
  • C ( βˆ’ 7 , βˆ’ 2 )
  • D ( 2 , βˆ’ 7 )

Q9:

Find the vertex of the graph of 𝑦 = βˆ’ π‘₯ βˆ’ 4 2 .

  • A ( βˆ’ 4 , 0 )
  • B ( 0 , 4 )
  • C ( 4 , 0 )
  • D ( 0 , βˆ’ 4 )
  • E ( βˆ’ 4 , βˆ’ 4 )

Q10:

Consider the curve shown below.

Which suitable triple 𝐴 , 𝐡 , 𝐢 would make this the graph of 𝑓 ( π‘₯ ) = 𝐴 ( π‘₯ βˆ’ 𝐡 ) + 𝐢 2 ?

  • A 1 4 , βˆ’ 4 , 7
  • B 1 4 , 4 , βˆ’ 7
  • C 1 2 , 4 , 7
  • D 1 4 , βˆ’ 4 , βˆ’ 7
  • E 1 2 , βˆ’ 4 , 7

Q11:

Find the vertex of the graph of 𝑦 = βˆ’ ( π‘₯ + 2 ) 2 .

  • A ( 0 , βˆ’ 2 )
  • B ( 2 , 0 )
  • C ( 0 , 2 )
  • D ( βˆ’ 2 , 0 )
  • E ( βˆ’ 2 , βˆ’ 2 )

Q12:

Find the vertex of the quadratic equation 𝑦 = βˆ’ 2 π‘₯ + 1 2 π‘₯ βˆ’ 1 2 .

  • A ( 2 , 1 5 )
  • B ( 6 , βˆ’ 1 )
  • C ( 1 , 9 )
  • D ( 3 , 1 7 )
  • E ( 4 , 1 5 )

Q13:

Suppose that 𝑓 ( π‘₯ ) = 𝐴 ( π‘₯ βˆ’ 1 ) ( π‘₯ βˆ’ 4 ) for some constant 𝐴 . What is the 𝑦 -coordinate of the vertex of parabola 𝑦 = 𝑓 ( π‘₯ ) ?

  • A βˆ’ 5 𝐴 2
  • B 9 𝐴 4
  • C βˆ’ 5 2
  • D βˆ’ 9 𝐴 4
  • E 9 4

Q14:

A parabola has the equation 𝑦 = 7 π‘₯ 2 .

What are the coordinates of its focus?

  • A ( 7 , 0 )
  • B ο€Ό 0 , 7 4 
  • C ( 0 , 7 )
  • D ο€Ό 7 4 , 0 
  • E ο€Ό 7 2 , 0 

Write an equation for its directrix.

  • A π‘₯ + 7 4 = 0
  • B π‘₯ βˆ’ 7 = 0
  • C π‘₯ + 7 = 0
  • D π‘₯ βˆ’ 7 4 = 0
  • E π‘₯ + 7 2 = 0

Q15:

Write an equation for the parabola whose focus is the point ο€Ώ βˆ’ √ 5 6 , 0  and whose directrix is the line π‘₯ = √ 5 6 .

  • A 𝑦 = βˆ’ √ 5 6 2
  • B 𝑦 = 2 √ 5 3 2
  • C 𝑦 = √ 5 6 2
  • D 𝑦 = βˆ’ 2 √ 5 3 2
  • E 𝑦 = βˆ’ √ 5 2 4 2

Q16:

Consider a parabola described by the parametric equations π‘₯ = 1 1 𝑑 2 , 𝑦 = 2 2 𝑑 , 𝑑 ∈ ℝ .

What are the coordinates of the focus?

  • A ( 0 , 2 2 )
  • B ( 0 , 1 1 )
  • C ( 2 2 , 1 1 )
  • D ( 1 1 , 0 )
  • E ( 1 1 , 2 2 )

What is the equation of the directrix?

  • A π‘₯ + 1 1 = 0
  • B 𝑦 + 1 1 = 0
  • C 𝑦 + 2 2 = 0
  • D π‘₯ βˆ’ 1 1 = 0
  • E 𝑦 βˆ’ 2 2 = 0

Q17:

The figure shows the curve ( 𝑦 + π‘₯ βˆ’ 3 ) + 4 ( 𝑦 βˆ’ π‘₯ ) + 4 = 0  , the dashed line 𝑦 + π‘₯ = 3 , and its perpendicular 𝑦 βˆ’ π‘₯ = βˆ’ 5 .

Determine the coordinates of the two intersections 𝐴 and 𝐡 .

  • A 𝐴 = ( 4 , βˆ’ 1 ) , 𝐡 = ( 1 , 2 )
  • B 𝐴 = ( 3 , βˆ’ 2 ) , 𝐡 = ( 1 , 6 )
  • C 𝐴 = ( 1 , βˆ’ 4 ) , 𝐡 = ( 2 , 1 )
  • D 𝐴 = ( 2 , βˆ’ 3 ) , 𝐡 = ( 6 , 1 )
  • E 𝐴 = ( 2 , βˆ’ 3 ) , 𝐡 = ( 4 , 1 )