Lesson Worksheet: Quadratic Functions in Different Forms Mathematics

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

Q1:

Rewrite the expression 𝑥+14𝑥 in the form (𝑥+𝑝)+𝑞.

  • A(𝑥+7)49
  • B(𝑥14)+196
  • C(𝑥7)+49
  • D(𝑥7)49
  • E(𝑥+14)196

What is the minimum value of the function 𝑓(𝑥)=𝑥+14𝑥?

Q2:

Find the vertex of the graph of 𝑦=𝑥.

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

Q3:

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

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

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?

  • A𝑓(𝑥)=2(𝑥3)+8
  • B𝑓(𝑥)=2(𝑥+3)4
  • C𝑓(𝑥)=2(𝑥+3)+8
  • D𝑓(𝑥)=2(𝑥+3)8
  • E𝑓(𝑥)=2(𝑥3)+4

Q5:

Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at (3,17)
  • 𝑓(4)=5
  • A𝑓(𝑥)=22(𝑥3)+17
  • B𝑓(𝑥)=22(𝑥3)17
  • C𝑓(𝑥)=17(𝑥3)17
  • DThe function does not exist.
  • E𝑓(𝑥)=22(𝑥+3)17

Q6:

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

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

Q7:

Consider the function 𝑓(𝑥)=𝑎𝑥+𝑏𝑥+𝑐 where 𝑎0. What is the 𝑥-coordinate of the vertex of its curve?

  • A𝑏2𝑎
  • B𝑏2𝑎
  • C𝑎2𝑏
  • D𝑎2𝑏

Q8:

Find the vertex of the graph of 𝑦=(𝑥3)+2.

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

Q9:

Rewrite the expression 4𝑥12𝑥+13 in the form 𝑎(𝑥+𝑝)+𝑞.

  • A4𝑥34+2
  • B4𝑥32+4
  • C4(𝑥+3)23
  • D4𝑥+32+4
  • E4(𝑥3)23

What is the minimum value of the function 𝑓(𝑥)=4𝑥12𝑥+13?

Q10:

Rewrite the expression 𝑥12𝑥+20 in the form (𝑥+𝑝)+𝑞.

  • A(𝑥6)16
  • B(𝑥12)+20
  • C(𝑥+6)16
  • D(𝑥12)20
  • E(𝑥6)+16

What is the minimum value of the function 𝑓(𝑥)=𝑥12𝑥+20?

Q11:

Rewrite the expression 4𝑥8𝑥1 in the form 𝑎(𝑥+𝑝)+𝑞.

  • A4(𝑥+1)+3
  • B4(𝑥+1)+5
  • C4(𝑥1)+3
  • D4(𝑥1)5
  • E4(𝑥+1)3

What is the maximum value of the function 𝑓(𝑥)=4𝑥8𝑥1?

Q12:

In completing the square for quadratic function 𝑓(𝑥)=𝑥+14𝑥+46, you arrive at the expression (𝑥𝑏)+𝑐. What is the value of 𝑏?

Q13:

Which of the following is the vertex form of the function 𝑓(𝑥)=2𝑥+12𝑥+11?

  • A𝑓(𝑥)=2(𝑥+3)7
  • B𝑓(𝑥)=(2𝑥+3)7
  • C𝑓(𝑥)=(2𝑥3)7
  • D𝑓(𝑥)=2(𝑥3)7
  • E𝑓(𝑥)=(𝑥+3)7

Q14:

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

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

Q15:

Find the vertex of the graph of 𝑦=𝑥.

  • A(0,1)
  • B(0,0)
  • C(1,1)
  • 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 𝑎=23𝑙𝑧, find the area of the figure included between the 𝑥-axis and the curve of the quadratic function 𝑓(𝑥)=𝑥12𝑥+32 in square units.

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

Q17:

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

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

Q18:

In the figure below, the area included between the curve of the quadratic function 𝑓(𝑥)=𝑥16𝑥+55 and the line segment 𝑙 lying on the 𝑥-axis is calculated by the relation 𝑎=23𝑙𝑧. Represent the function 𝑔(𝑥)=|𝑥8|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(𝑥)=𝑥+𝑚. Find the area of triangle 𝐴𝐵𝐶 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 𝑎, the age of the younger sibling.

  • A𝑃=𝑎(𝑎+2)
  • B𝑃=𝑎(𝑎+3)
  • C𝑃=3𝑎
  • D𝑃=𝑎(𝑎2)
  • E𝑃=𝑎(𝑎3)

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