Worksheet: Quadratic Functions in Different Forms

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

Q1:

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

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

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

Q2:

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 )

Q3:

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 )

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

Q5:

Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at (3,βˆ’17)
  • 𝑓 ( 4 ) = 5
  • A 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ βˆ’ 3 ) + 1 7 
  • B 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ βˆ’ 3 ) βˆ’ 1 7 
  • C 𝑓 ( π‘₯ ) = 1 7 ( π‘₯ βˆ’ 3 ) βˆ’ 1 7 
  • DThe function does not exist.
  • E 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ + 3 ) βˆ’ 1 7 

Q6:

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

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

Q7:

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?

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

Q8:

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 )

Q9:

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

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

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

Q10:

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

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

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

Q11:

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

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

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

Q12:

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?

Q13:

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

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

Q14:

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 )

Q15:

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 )

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=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
  • B 3 2 3 square units
  • C 6 4 3 square units
  • D 8 3 square units

Q17:

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 )

Q18:

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

Q19:

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

Q20:

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.

  • A 𝑃 = π‘Ž ( π‘Ž + 2 )
  • B 𝑃 = π‘Ž ( π‘Ž + 3 )
  • C 𝑃 = 3 π‘Ž 
  • D 𝑃 = π‘Ž ( π‘Ž βˆ’ 2 )
  • E 𝑃 = π‘Ž ( π‘Ž βˆ’ 3 )

Q21:

A cell phone company has the following cost and revenue functions: 𝐢(π‘₯)=8π‘₯βˆ’600π‘₯+21,500 and 𝑅(π‘₯)=βˆ’3π‘₯+480π‘₯, where π‘₯ is the number of cell phones. State the range for the number of cell phones they can produce while making a profit. Round your answers to the nearest integer that guarantees a profit.

  • A28–70 cell phones
  • BMore than 160 cell phones
  • C28–71 cell phones
  • D27–70 cell phones

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