Worksheet: Quadratic Functions in Different Forms

In this worksheet, we will practice evaluating and writing a quadratic function in different forms.


Rewrite the expression π‘₯+14π‘₯ in the form (π‘₯+𝑝)+π‘žοŠ¨.

  • A(π‘₯+7)βˆ’49
  • B(π‘₯βˆ’14)+196
  • C(π‘₯βˆ’7)+49
  • D(π‘₯βˆ’7)βˆ’49
  • E(π‘₯+14)βˆ’196

What is the minimum value of the function 𝑓(π‘₯)=π‘₯+14π‘₯?


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

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


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

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


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?

  • A𝑓(π‘₯)=2(π‘₯βˆ’3)+8
  • B𝑓(π‘₯)=βˆ’2(π‘₯+3)βˆ’4
  • C𝑓(π‘₯)=βˆ’2(π‘₯+3)+8
  • D𝑓(π‘₯)=2(π‘₯+3)βˆ’8
  • E𝑓(π‘₯)=βˆ’2(π‘₯βˆ’3)+4


Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at (3,βˆ’17)
  • 𝑓(4)=5
  • A𝑓(π‘₯)=22(π‘₯βˆ’3)+17
  • B𝑓(π‘₯)=22(π‘₯βˆ’3)βˆ’17
  • C𝑓(π‘₯)=17(π‘₯βˆ’3)βˆ’17
  • DThe function does not exist.
  • E𝑓(π‘₯)=22(π‘₯+3)βˆ’17


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

  • A𝐴=20
  • B𝐴=βˆ’20
  • C𝐴=βˆ’13
  • D𝐴=βˆ’33
  • E𝐴=13


Consider the function 𝑓(π‘₯)=π‘Žπ‘₯+𝑏π‘₯+π‘οŠ¨ where π‘Žβ‰ 0. What is the π‘₯-coordinate of the vertex of its curve?

  • A𝑏2π‘Ž
  • Bβˆ’π‘2π‘Ž
  • Cβˆ’π‘Ž2𝑏
  • Dπ‘Ž2𝑏


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

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


Rewrite the expression 4π‘₯βˆ’12π‘₯+13 in the form π‘Ž(π‘₯+𝑝)+π‘žοŠ¨.

  • A4ο€Όπ‘₯βˆ’34+2
  • B4ο€Όπ‘₯βˆ’32+4
  • C4(π‘₯+3)βˆ’23
  • D4ο€Όπ‘₯+32+4
  • E4(π‘₯βˆ’3)βˆ’23

What is the minimum value of the function 𝑓(π‘₯)=4π‘₯βˆ’12π‘₯+13?


Rewrite the expression π‘₯βˆ’12π‘₯+20 in the form (π‘₯+𝑝)+π‘žοŠ¨.

  • A(π‘₯βˆ’6)βˆ’16
  • B(π‘₯βˆ’12)+20
  • C(π‘₯+6)βˆ’16
  • D(π‘₯βˆ’12)βˆ’20
  • E(π‘₯βˆ’6)+16

What is the minimum value of the function 𝑓(π‘₯)=π‘₯βˆ’12π‘₯+20?


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

  • Aβˆ’4(π‘₯+1)+3
  • Bβˆ’4(π‘₯+1)+5
  • Cβˆ’4(π‘₯βˆ’1)+3
  • D4(π‘₯βˆ’1)βˆ’5
  • E4(π‘₯+1)βˆ’3

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


In completing the square for quadratic function 𝑓(π‘₯)=π‘₯+14π‘₯+46, you arrive at the expression (π‘₯βˆ’π‘)+π‘οŠ¨. What is the value of 𝑏?


Which of the following is the vertex form of the function 𝑓(π‘₯)=2π‘₯+12π‘₯+11?

  • A𝑓(π‘₯)=2(π‘₯+3)βˆ’7
  • B𝑓(π‘₯)=(2π‘₯+3)βˆ’7
  • C𝑓(π‘₯)=(2π‘₯βˆ’3)βˆ’7
  • D𝑓(π‘₯)=2(π‘₯βˆ’3)βˆ’7
  • E𝑓(π‘₯)=(π‘₯+3)βˆ’7


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

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


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

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


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=23𝑙𝑧, find the area of the figure included between the π‘₯-axis and the curve of the quadratic function 𝑓(π‘₯)=π‘₯βˆ’12π‘₯+32 in square units.

  • A32 square units
  • B323 square units
  • C643 square units
  • D83 square units


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

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


In the figure below, the area included between the curve of the quadratic function 𝑓(π‘₯)=π‘₯βˆ’16π‘₯+55 and the line segment 𝑙 lying on the π‘₯-axis is calculated by the relation π‘Ž=23𝑙𝑧. 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.

  • A23 square units
  • B13 square units
  • C27 square units
  • D5 square units


The figure below represents the function 𝑓(π‘₯)=π‘₯+π‘šοŠ¨. Find the area of triangle 𝐴𝐡𝐢 given 𝑂𝐴=9.


Two siblings are 3 years apart in age. Write an equation for 𝑃, the product of their ages, in terms of π‘Ž, the age of the younger sibling.

  • A𝑃=π‘Ž(π‘Ž+2)
  • B𝑃=π‘Ž(π‘Ž+3)
  • C𝑃=3π‘ŽοŠ¨
  • D𝑃=π‘Ž(π‘Žβˆ’2)
  • E𝑃=π‘Ž(π‘Žβˆ’3)

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