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(𝑥−14)+196
  • C(𝑥−7)+49
  • D(𝑥−7)−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(𝑥−3)+8
  • B𝑓(𝑥)=−2(𝑥+3)−4
  • C𝑓(𝑥)=−2(𝑥+3)+8
  • D𝑓(𝑥)=2(𝑥+3)−8
  • E𝑓(𝑥)=−2(𝑥−3)+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(𝑥−3)+17
  • B𝑓(𝑥)=22(𝑥−3)−17
  • C𝑓(𝑥)=17(𝑥−3)−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 𝑎≠0. What is the 𝑥-coordinate of the vertex of its curve?

  • A𝑏2𝑎
  • B−𝑏2𝑎
  • C−𝑎2𝑏
  • D𝑎2𝑏

Q8:

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

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

Q9:

Rewrite the expression 4𝑥−12𝑥+13 in the form 𝑎(𝑥+𝑝)+𝑞.

  • A4𝑥−34+2
  • B4𝑥−32+4
  • C4(𝑥+3)−23
  • D4𝑥+32+4
  • E4(𝑥−3)−23

What is the minimum value of the function 𝑓(𝑥)=4𝑥−12𝑥+13?

Q10:

Rewrite the expression 𝑥−12𝑥+20 in the form (𝑥+𝑝)+𝑞.

  • A(𝑥−6)−16
  • B(𝑥−12)+20
  • C(𝑥+6)−16
  • D(𝑥−12)−20
  • E(𝑥−6)+16

What is the minimum value of the function 𝑓(𝑥)=𝑥−12𝑥+20?

Q11:

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

  • A−4(𝑥+1)+3
  • B−4(𝑥+1)+5
  • C−4(𝑥−1)+3
  • D4(𝑥−1)−5
  • E4(𝑥+1)−3

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

Q12:

In completing the square for quadratic function 𝑓(𝑥)=𝑥+14𝑥+46, you arrive at the expression (𝑥−𝑏)+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)=(2𝑥−3)−7
  • D𝑓(𝑥)=2(𝑥−3)−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)=𝑥−12𝑥+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−(−4−𝑥).

  • 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)=𝑥−16𝑥+55 and the line segment 𝑙 lying on the 𝑥-axis is calculated by the relation 𝑎=23𝑙𝑧. 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.

  • 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(𝑎−2)
  • E𝑃=𝑎(𝑎−3)

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