Worksheet: Properties of Parabolas

In this worksheet, we will practice identifying the properties of the parabola using graphical representation and the function rule.

Q1:

Find the coordinates of the vertex of the function 𝑓 ( 𝑥 ) = 8 𝑥 6 + 4 𝑥 2 .

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

Q2:

Find the vertex of the graph of 𝑦 = ( 3 𝑥 + 1 ) 2 .

  • A 0 , 1 3
  • B 1 3 , 0
  • C 0 , 1 3
  • D 1 3 , 0
  • E 1 3 , 1 3

Q3:

Find the vertex of the graph of 𝑦 = ( 𝑥 1 ) 2 .

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

Q4:

Find the vertex of the graph of 𝑦 = ( 𝑥 + 2 ) 2 .

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

Q5:

Find the vertex of the graph of 𝑦 = 𝑥 4 2 .

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

Q6:

Find the point of symmetry of the curve of the function 𝑓 ( 𝑥 ) = 7 ( 𝑥 2 ) 2 .

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

Q7:

Find the vertex of the quadratic equation 𝑦 = 2 𝑥 + 1 2 𝑥 1 2 .

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

Q8:

Consider the curve shown below.

Which suitable triple 𝐴 , 𝐵 , 𝐶 would make this the graph of 𝑓 ( 𝑥 ) = 𝐴 ( 𝑥 𝐵 ) + 𝐶 2 ?

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

Q9:

The figure shows the curve ( 𝑦 + 𝑥 3 ) + 4 ( 𝑦 𝑥 ) + 4 = 0 , the dashed line 𝑦 + 𝑥 = 3 , and its perpendicular 𝑦 𝑥 = 5 .

Determine the coordinates of the two intersections 𝐴 and 𝐵 .

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

Q10:

The figure shows the curve ( 𝑦 + 𝑥 3 ) + 4 ( 𝑦 𝑥 ) + 4 = 0 2 , the dashed line 𝑦 + 𝑥 = 3 , and its perpendicular 𝑦 𝑥 = 5 .

Determine the coordinates of the two intersections 𝐴 and 𝐵 .

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

The vertex of this parabola lies where the line 𝑦 𝑥 = 𝐾 meets the parabola at exactly one point.

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

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

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

By completing the square and rewriting this as ( 𝑦 + 𝑥 𝐴 ) + 4 ( 𝑦 𝑥 ) + 𝐵 = 0 2 , determine what the new axis of symmetry is. What is the value of the constant 𝐵 ?

  • A 𝑦 + 𝑥 = 1 , 𝐵 = 2
  • B 𝑦 + 𝑥 = 4 , 𝐵 = 1 4
  • C 𝑦 + 𝑥 = 7 , 𝐵 = 1 4
  • D 𝑦 + 𝑥 = 1 , 𝐵 = 3
  • E 𝑦 + 𝑥 = 4 , 𝐵 = 2

By determining 𝐴 , 𝐵 such that 𝑦 5 𝑥 = 𝐴 ( 𝑦 + 𝑥 ) + 𝐵 ( 𝑦 𝑥 ) , complete the squares and rewrite ( 𝑦 + 𝑥 ) + 𝑦 5 𝑥 + 7 2 as ( 𝑦 + 𝑥 𝑎 ) + 𝑏 ( 𝑦 𝑥 ) + 𝑐 2 . What is this expression? What is the vertex of the parabola?

  • A ( 𝑦 + 𝑥 1 ) + 3 ( 𝑦 𝑥 ) + 6 2 , 3 2 , 1 2
  • B ( 𝑦 + 𝑥 1 ) + 3 ( 𝑦 𝑥 ) + 7 2 , 3 2 , 1 2
  • C ( 𝑦 + 𝑥 + 1 ) + 3 ( 𝑦 𝑥 ) + 6 2 , 3 2 , 1 2
  • D ( 𝑦 + 𝑥 + 1 ) 3 ( 𝑦 𝑥 ) + 6 2 , 3 2 , 1 2
  • E ( 𝑦 + 𝑥 1 ) 3 ( 𝑦 𝑥 ) + 6 2 , 3 2 , 1 2

Q11:

Consider the parabola with Cartesian equation 𝑦 = 2 1 7 𝑥 .

What are the coordinates of its focus?

  • A 2 1 7 , 0
  • B 0 , 1 7 2
  • C 0 , 2 1 7
  • D 1 7 2 , 0
  • E 0 , 1 7 2

Write the equation of its directrix.

  • A 𝑥 = 1 7 2
  • B 𝑥 = 2 1 7
  • C 𝑥 = 2 1 7
  • D 𝑥 = 1 7 2
  • E 𝑥 = 8 1 7

Q12:

A parabola has the equation 𝑦 = 7 𝑥 2 .

What are the coordinates of its focus?

  • A ( 7 , 0 )
  • B 0 , 7 4
  • C ( 0 , 7 )
  • D 7 4 , 0
  • E 7 2 , 0

Write an equation for its directrix.

  • A 𝑥 + 7 4 = 0
  • B 𝑥 7 = 0
  • C 𝑥 + 7 = 0
  • D 𝑥 7 4 = 0
  • E 𝑥 + 7 2 = 0

Q13:

Consider a parabola described by the parametric equations 𝑥 = 1 1 𝑡 2 , 𝑦 = 2 2 𝑡 , 𝑡 .

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 𝑥 + 1 1 = 0
  • B 𝑦 + 1 1 = 0
  • C 𝑦 + 2 2 = 0
  • D 𝑥 1 1 = 0
  • E 𝑦 2 2 = 0

Q14:

A parabola has an equation ( 𝑥 4 ) = 2 0 ( 𝑦 + 3 ) .

Find the coordinates of the vertex.

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

Determine the equation of the directrix.

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

Q15:

Consider the parabola with the Cartesian equation 𝑥 = 6 1 1 𝑦 2 .

What are the coordinates of the focus of the parabola with the Cartesian equation 𝑥 = 6 1 1 𝑦 2 ?

  • A 0 , 6 1 1
  • B 3 1 1 2 , 0
  • C 6 1 1 , 0
  • D 0 , 3 1 1 2
  • E 0 , 2 4 1 1

Write the equation of its directrix.

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

Q16:

Write an equation for the parabola whose focus is the point 5 6 , 0 and whose directrix is the line 𝑥 = 5 6 .

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

Q17:

Suppose that 𝑓 ( 𝑥 ) = 𝐴 ( 𝑥 1 ) ( 𝑥 4 ) for some constant 𝐴 . What is the 𝑦 -coordinate of the vertex of parabola 𝑦 = 𝑓 ( 𝑥 ) ?

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

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