Worksheet: Writing Quadratic Functions in Vertex Form

Practice

In this worksheet, we will practice writing a quadratic function in vertex form.

Q1:

Rewrite the expression 𝑥 + 1 4 𝑥 2 in the form ( 𝑥 + 𝑝 ) + 𝑞 2 .

  • A ( 𝑥 − 7 ) + 4 9 2
  • B ( 𝑥 − 7 ) − 4 9 2
  • C ( 𝑥 + 1 4 ) − 1 9 6 2
  • D ( 𝑥 + 7 ) − 4 9 2
  • E ( 𝑥 − 1 4 ) + 1 9 6 2

What is the minimum value of the function 𝑓 ( 𝑥 ) = 𝑥 + 1 4 𝑥 2 ?

Q2:

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

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

Q3:

Find the vertex of the graph of 𝑦 = 5 ( 𝑥 + 1 ) + 6 2 .

  • A ( 6 , − 1 )
  • B ( 1 , 6 )
  • C ( 6 , 1 )
  • D ( − 1 , 6 )
  • E ( − 1 , − 6 )

Q4:

Consider the graph:

Which is the following is the same as the function 𝑓 ( 𝑥 ) = − 2 ( 𝑥 + 1 ) ( 𝑥 + 5 ) whose graph is shown?

  • A 𝑓 ( 𝑥 ) = − 2 ( 𝑥 + 3 ) − 4 2
  • B 𝑓 ( 𝑥 ) = 2 ( 𝑥 + 3 ) − 8 2
  • C 𝑓 ( 𝑥 ) = 2 ( 𝑥 − 3 ) + 8 2
  • D 𝑓 ( 𝑥 ) = − 2 ( 𝑥 + 3 ) + 8 2
  • E 𝑓 ( 𝑥 ) = − 2 ( 𝑥 − 3 ) + 4 2

Q5:

Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at ( 3 , − 1 7 )
  • 𝑓 ( 4 ) = 5
  • A 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 + 3 ) − 1 7 2
  • B The function does not exist.
  • C 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 − 3 ) + 1 7 2
  • D 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 − 3 ) − 1 7 2
  • E 𝑓 ( 𝑥 ) = 1 7 ( 𝑥 − 3 ) − 1 7 2

Q6:

By writing 𝑓 ( 𝑥 ) = − 𝑥 + 8 𝑥 + 𝐴 2 in vertex form, find 𝐴 such that 𝑓 ( 𝑥 ) = 3 has exactly one solution.

  • A 𝐴 = − 2 0
  • B 𝐴 = 1 3
  • C 𝐴 = 2 0
  • D 𝐴 = − 1 3
  • E 𝐴 = − 3 3

Q7:

Consider the function 𝑓 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 2 where 𝑎 ≠ 0 . What is the 𝑥 -coordinate of the vertex of its curve?

  • A − 𝑎 2 𝑏
  • B 𝑏 2 𝑎
  • C 𝑎 2 𝑏
  • D − 𝑏 2 𝑎

Q8:

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

  • A ( 2 , 3 )
  • B ( − 3 , 2 )
  • C ( 2 , − 3 )
  • D ( 3 , 2 )
  • E ( − 2 , 3 )

Q9:

Rewrite the expression 4 𝑥 − 1 2 𝑥 + 1 3 2 in the form 𝑎 ( 𝑥 + 𝑝 ) + 𝑞 2 .

  • A 4 𝑥 + 3 2 + 4 2
  • B 4 ( 𝑥 − 3 ) − 2 3 2
  • C 4 ( 𝑥 + 3 ) − 2 3 2
  • D 4 𝑥 − 3 2 + 4 2
  • E 4 𝑥 − 3 4 + 2 2

What is the minimum value of the function 𝑓 ( 𝑥 ) = 4 𝑥 − 1 2 𝑥 + 1 3 2 ?

Q10:

Rewrite the expression 𝑥 − 1 2 𝑥 + 2 0 in the form ( 𝑥 + 𝑝 ) + 𝑞 .

  • A ( 𝑥 − 6 ) + 1 6
  • B ( 𝑥 − 1 2 ) + 2 0
  • C ( 𝑥 + 6 ) − 1 6
  • D ( 𝑥 − 6 ) − 1 6
  • E ( 𝑥 − 1 2 ) − 2 0

What is the minimum value of the function 𝑓 ( 𝑥 ) = 𝑥 − 1 2 𝑥 + 2 0 ?

Q11:

Rewrite the expression − 4 𝑥 − 8 𝑥 − 1 2 in the form 𝑎 ( 𝑥 + 𝑝 ) + 𝑞 2 .

  • A 4 ( 𝑥 − 1 ) − 5 2
  • B 4 ( 𝑥 + 1 ) − 3 2
  • C − 4 ( 𝑥 − 1 ) + 3 2
  • D − 4 ( 𝑥 + 1 ) + 3 2
  • E − 4 ( 𝑥 + 1 ) + 5 2

What is the maximum value of the function 𝑓 ( 𝑥 ) = − 4 𝑥 − 8 𝑥 − 1 2 ?

Q12:

In completing the square for quadratic function 𝑓 ( 𝑥 ) = 𝑥 + 1 4 𝑥 + 4 6 2 , you arrive at the expression ( 𝑥 − 𝑏 ) + 𝑐 2 . What is the value of 𝑏 ?

Q13:

Which of the following is the vertex form of the function 𝑓 ( 𝑥 ) = 2 𝑥 + 1 2 𝑥 + 1 1 2 ?

  • A 𝑓 ( 𝑥 ) = 2 ( 𝑥 − 3 ) − 7 2
  • B 𝑓 ( 𝑥 ) = ( 2 𝑥 + 3 ) − 7 2
  • C 𝑓 ( 𝑥 ) = ( 2 𝑥 − 3 ) − 7 2
  • D 𝑓 ( 𝑥 ) = 2 ( 𝑥 + 3 ) − 7 2
  • E 𝑓 ( 𝑥 ) = ( 𝑥 + 3 ) − 7 2

Q14:

Find the vertex of the graph of 𝑦 = 𝑥 + 7 2 .

  • A ( 7 , 0 )
  • B ( 0 , − 7 )
  • C ( − 7 , 0 )
  • D ( 0 , 7 )
  • E ( 7 , 7 )

Q15:

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

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

Q16:

If the area included between the curve of a quadratic function and a horizontal line segment joining any two points lying on it, as shown in the figure below, is calculated by the relation 𝑎 = 2 3 𝑙 𝑧 , find the area of the figure included between the 𝑥 -axis and the curve of the quadratic function 𝑓 ( 𝑥 ) = 𝑥 − 1 2 𝑥 + 3 2 2 in square units.

  • A 6 4 3 square units
  • B32 square units
  • C 8 3 square units
  • D 3 2 3 square units

Q17:

Find the coordinates of the vertex of the curve 𝑓 ( 𝑥 ) = 8 − ( − 4 − 𝑥 ) 2 .

  • A ( 4 , 8 )
  • B ( 8 , 4 )
  • C ( 8 , − 4 )
  • D ( − 4 , 8 )
  • E ( 4 , − 8 )

Q18:

In the figure below, the area included between the curve of the quadratic function 𝑓 ( 𝑥 ) = 𝑥 − 1 6 𝑥 + 5 5 2 and the line segment 𝑙 lying on the 𝑥 -axis is calculated by the relation 𝑎 = 2 3 𝑙 𝑧 . Represent the function 𝑔 ( 𝑥 ) = | 𝑥 − 8 | − 3 on the same lattice to find the area of the part included between the two functions in area units.

  • A13 square units
  • B23 square units
  • C5 square units
  • D27 square units

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