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Lesson: Writing Quadratic Functions in Vertex Form

Sample Question Videos

Worksheet • 18 Questions • 1 Video

Q1:

Rewrite the expression π‘₯ + 1 4 π‘₯ 2 in the form ( π‘₯ + 𝑝 ) + π‘ž 2 .

  • A ( π‘₯ + 7 ) βˆ’ 4 9 2
  • B ( π‘₯ βˆ’ 1 4 ) + 1 9 6 2
  • C ( π‘₯ βˆ’ 7 ) βˆ’ 4 9 2
  • D ( π‘₯ + 1 4 ) βˆ’ 1 9 6 2
  • E ( π‘₯ βˆ’ 7 ) + 4 9 2

What is the minimum value of the function 𝑓 ( π‘₯ ) = π‘₯ + 1 4 π‘₯ 2 ?

Q2:

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

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

Q3:

Find the vertex of the graph of 𝑦 = 5 ( π‘₯ + 1 ) + 6 2 .

  • A ( βˆ’ 1 , 6 )
  • B ( βˆ’ 1 , βˆ’ 6 )
  • C ( 1 , 6 )
  • D ( 6 , 1 )
  • E ( 6 , βˆ’ 1 )

Q4:

Consider the graph:

Which is the following is the same as the function 𝑓 ( π‘₯ ) = βˆ’ 2 ( π‘₯ + 1 ) ( π‘₯ + 5 ) whose graph is shown?

  • A 𝑓 ( π‘₯ ) = βˆ’ 2 ( π‘₯ + 3 ) + 8 2
  • B 𝑓 ( π‘₯ ) = βˆ’ 2 ( π‘₯ βˆ’ 3 ) + 4 2
  • C 𝑓 ( π‘₯ ) = 2 ( π‘₯ + 3 ) βˆ’ 8 2
  • D 𝑓 ( π‘₯ ) = 2 ( π‘₯ βˆ’ 3 ) + 8 2
  • E 𝑓 ( π‘₯ ) = βˆ’ 2 ( π‘₯ + 3 ) βˆ’ 4 2

Q5:

Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at ( 3 , βˆ’ 1 7 )
  • 𝑓 ( 4 ) = 5
  • A 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ βˆ’ 3 ) βˆ’ 1 7 2
  • B 𝑓 ( π‘₯ ) = 1 7 ( π‘₯ βˆ’ 3 ) βˆ’ 1 7 2
  • C The function does not exist.
  • D 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ βˆ’ 3 ) + 1 7 2
  • E 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ + 3 ) βˆ’ 1 7 2

Q6:

By writing 𝑓 ( π‘₯ ) = βˆ’ π‘₯ + 8 π‘₯ + 𝐴 2 in vertex form, find 𝐴 such that 𝑓 ( π‘₯ ) = 3 has exactly one solution.

  • A 𝐴 = βˆ’ 1 3
  • B 𝐴 = βˆ’ 3 3
  • C 𝐴 = 1 3
  • D 𝐴 = 2 0
  • E 𝐴 = βˆ’ 2 0

Q7:

Consider the function 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 2 where π‘Ž β‰  0 . What is the π‘₯ -coordinate of the vertex of its curve?

  • A βˆ’ 𝑏 2 π‘Ž
  • B 𝑏 2 π‘Ž
  • C π‘Ž 2 𝑏
  • D βˆ’ π‘Ž 2 𝑏

Q8:

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

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

Q9:

Rewrite the expression 4 π‘₯ βˆ’ 1 2 π‘₯ + 1 3 2 in the form π‘Ž ( π‘₯ + 𝑝 ) + π‘ž 2 .

  • A 4 ο€Ό π‘₯ βˆ’ 3 2  + 4 2
  • B 4 ο€Ό π‘₯ βˆ’ 3 4  + 2 2
  • C 4 ( π‘₯ βˆ’ 3 ) βˆ’ 2 3 2
  • D 4 ( π‘₯ + 3 ) βˆ’ 2 3 2
  • E 4 ο€Ό π‘₯ + 3 2  + 4 2

What is the minimum value of the function 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 1 2 π‘₯ + 1 3 2 ?

Q10:

Rewrite the expression π‘₯ βˆ’ 1 2 π‘₯ + 2 0  in the form ( π‘₯ + 𝑝 ) + π‘ž  .

  • A ( π‘₯ βˆ’ 6 ) βˆ’ 1 6 
  • B ( π‘₯ βˆ’ 1 2 ) βˆ’ 2 0 
  • C ( π‘₯ βˆ’ 1 2 ) + 2 0 
  • D ( π‘₯ + 6 ) βˆ’ 1 6 
  • E ( π‘₯ βˆ’ 6 ) + 1 6 

What is the minimum value of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 2 π‘₯ + 2 0  ?

Q11:

Rewrite the expression βˆ’ 4 π‘₯ βˆ’ 8 π‘₯ βˆ’ 1 2 in the form π‘Ž ( π‘₯ + 𝑝 ) + π‘ž 2 .

  • A βˆ’ 4 ( π‘₯ + 1 ) + 3 2
  • B βˆ’ 4 ( π‘₯ + 1 ) + 5 2
  • C 4 ( π‘₯ + 1 ) βˆ’ 3 2
  • D βˆ’ 4 ( π‘₯ βˆ’ 1 ) + 3 2
  • E 4 ( π‘₯ βˆ’ 1 ) βˆ’ 5 2

What is the maximum value of the function 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 8 π‘₯ βˆ’ 1 2 ?

Q12:

In completing the square for quadratic function 𝑓 ( π‘₯ ) = π‘₯ + 1 4 π‘₯ + 4 6 2 , you arrive at the expression ( π‘₯ βˆ’ 𝑏 ) + 𝑐 2 . What is the value of 𝑏 ?

Q13:

Which of the following is the vertex form of the function 𝑓 ( π‘₯ ) = 2 π‘₯ + 1 2 π‘₯ + 1 1 2 ?

  • A 𝑓 ( π‘₯ ) = 2 ( π‘₯ + 3 ) βˆ’ 7 2
  • B 𝑓 ( π‘₯ ) = ( π‘₯ + 3 ) βˆ’ 7 2
  • C 𝑓 ( π‘₯ ) = ( 2 π‘₯ + 3 ) βˆ’ 7 2
  • D 𝑓 ( π‘₯ ) = ( 2 π‘₯ βˆ’ 3 ) βˆ’ 7 2
  • E 𝑓 ( π‘₯ ) = 2 ( π‘₯ βˆ’ 3 ) βˆ’ 7 2

Q14:

Find the vertex of the graph of 𝑦 = π‘₯ + 7 2 .

  • A ( 0 , 7 )
  • B ( 7 , 7 )
  • C ( 0 , βˆ’ 7 )
  • D ( βˆ’ 7 , 0 )
  • E ( 7 , 0 )

Q15:

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

  • A ( 0 , 0 )
  • B ( βˆ’ 1 , βˆ’ 1 )
  • C ( 1 , 1 )
  • D ( 1 , 0 )
  • E ( 0 , 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 π‘Ž = 2 3 𝑙 𝑧 , find the area of the figure included between the π‘₯ -axis and the curve of the quadratic function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 2 π‘₯ + 3 2 2 in square units.

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

Q17:

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

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

Q18:

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

  • A27 square units
  • B23 square units
  • C5 square units
  • D13 square units
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