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Worksheet: Features of Quadratic Functions

In this worksheet, we will practice identifying features of quadratic equations, such as its vertex, maximum or minimum value, axis of symmetry, domain, and range.

Q1:

Find the coordinates of the vertex of the graph of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 6 π‘₯ βˆ’ 4 2 . State the value of the function at the vertex and determine whether it is a maximum or minimum value.

  • A The vertex is ( βˆ’ 1 3 , 3 ) , the value of the function at the vertex is βˆ’ 1 3 , it is a minimum value.
  • B The vertex is ( 3 , βˆ’ 1 3 ) , the value of the function at the vertex is 3, it is a minimum value.
  • C The vertex is ( βˆ’ 1 3 , 3 ) , the value of the function at the vertex is 3, it is a maximum value.
  • D The vertex is ( 3 , βˆ’ 1 3 ) , the value of the function at the vertex is βˆ’ 1 3 , it is a minimum value.
  • E The vertex is ( 3 , βˆ’ 1 3 ) , the value of the function at the vertex is βˆ’ 1 3 , it is a maximum value.

Q2:

Find the coordinates of the vertex of the graph of 𝑓 ( π‘₯ ) = π‘₯ + 4 π‘₯ + 5 2 . State the value of the function at the vertex and determine whether it is a maximum or minimum value.

  • A The vertex is ( 1 , βˆ’ 2 ) , the value of the function at the vertex is 1, it is a minimum value.
  • B The vertex is ( βˆ’ 2 , 1 ) , the value of the function at the vertex is βˆ’ 2 , it is a minimum value.
  • C The vertex is ( 1 , βˆ’ 2 ) , the value of the function at the vertex is βˆ’ 2 , it is a maximum value.
  • D The vertex is ( βˆ’ 2 , 1 ) , the value of the function at the vertex is 1, it is a minimum value.
  • E The vertex is ( βˆ’ 2 , 1 ) , the value of the function at the vertex is 1, it is a maximum value.

Q3:

Find the coordinates of the vertex of the graph of 𝑓 ( π‘₯ ) = βˆ’ π‘₯ + 2 π‘₯ + 2 2 . State the value of the function at the vertex and determine whether it is a maximum or minimum value.

  • A The vertex is ( 3 , 1 ) , the value of the function at the vertex is 3, it is a maximum value.
  • B The vertex is ( 1 , 3 ) , the value of the function at the vertex is 1, it is a maximum value.
  • C The vertex is ( 3 , 1 ) , the value of the function at the vertex is 1, it is a minimum value.
  • D The vertex is ( 1 , 3 ) , the value of the function at the vertex is 3, it is a maximum value.
  • E The vertex is ( 1 , 3 ) , the value of the function at the vertex is 3, it is a minimum value.

Q4:

The following statements refer to the function 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐  . Which statement is true?

  • AThe range of the function is all positive real numbers.
  • BThe function has two roots.
  • CThe function has one relative maximum.
  • DThe function has a vertex at π‘₯ = βˆ’ 𝑏 2 π‘Ž , 𝑦 = 𝑓 ο€½ βˆ’ 𝑏 2 π‘Ž  .

Q5:

The graph of the function 𝑓 ( π‘₯ ) = π‘Ÿ π‘₯ + 𝑑 π‘₯ + 𝑧 2 passes through the point ( 0 , 0 ) . Given that the minimum value of the function is βˆ’ 8 , and the axis of symmetry is the line π‘₯ = 1 , find the the values of π‘Ÿ , 𝑑 , and 𝑧 .

  • A π‘Ÿ = 8 , 𝑑 = 1 6 , 𝑧 = 0
  • B π‘Ÿ = βˆ’ 8 , 𝑑 = 8 , 𝑧 = 0
  • C π‘Ÿ = 1 6 , 𝑑 = 8 , 𝑧 = 0
  • D π‘Ÿ = 8 , 𝑑 = βˆ’ 1 6 , 𝑧 = 0

Q6:

The graph of the quadratic function 𝑓 intersects the π‘₯ -axis at the points ( 2 , 0 ) and ( 4 , 0 ) . What is the π‘₯ -coordinate of the vertex of the graph?

Q7:

A ball is hurled upward from the top of a building. Its height in feet at time 𝑑 , in seconds, can be described by the function 𝐻 ( 𝑑 ) = βˆ’ 1 6 𝑑 + 6 4 𝑑 + 1 2 0  . How many seconds did it take for the ball to reach its maximum height?

Q8:

For the function 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + 5 π‘₯ + 2 1 2 , answer the following questions.

Find, by factoring, the zeros of the function.

  • A βˆ’ 3 , 7 4
  • B βˆ’ 3 , βˆ’ 7 4
  • C βˆ’ 7 , 3
  • D βˆ’ 7 4 , 3
  • E βˆ’ 3 , 7

Identify the graph of 𝑓 .

  • Athe red graph
  • Bthe yellow graph
  • Cthe blue graph

Write the equation for 𝑔 , the function that describes the yellow graph.

  • A 𝑔 ( π‘₯ ) = βˆ’ ο€Ή βˆ’ 4 π‘₯ βˆ’ 5 π‘₯ βˆ’ 2 1  2
  • B 𝑔 ( π‘₯ ) = βˆ’ 4 π‘₯ + 5 π‘₯ + 2 1 2
  • C 𝑔 ( π‘₯ ) = βˆ’ ο€Ή βˆ’ 4 π‘₯ + 5 π‘₯ + 2 1  2
  • D 𝑔 ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 5 π‘₯ + 2 1 2
  • E 𝑔 ( π‘₯ ) = βˆ’ ο€Ή βˆ’ 4 π‘₯ βˆ’ 5 π‘₯ + 2 1  2

Write the equation for β„Ž , the function that describes the blue graph.

  • A β„Ž ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 5 π‘₯ + 2 1 2
  • B β„Ž ( π‘₯ ) = βˆ’ 4 π‘₯ + 5 π‘₯ βˆ’ 2 1 2
  • C β„Ž ( π‘₯ ) = βˆ’ 4 π‘₯ + 5 π‘₯ + 2 1 2
  • D β„Ž ( π‘₯ ) = βˆ’ ο€Ή βˆ’ 4 π‘₯ βˆ’ 5 π‘₯ + 2 1  2
  • E β„Ž ( π‘₯ ) = βˆ’ ο€Ή βˆ’ 4 π‘₯ + 5 π‘₯ + 2 1  2

Q9:

Determine the domain and the range of the function .

  • A The domain is , and the range is .
  • B The domain is , and the range is .
  • C The domain is , and the range is .
  • D The domain is , and the range is .
  • E The domain is , and the range is .

Q10:

The function 𝑓 ( π‘₯ ) = π‘š βˆ’ 3 π‘₯ 2 intersects the π‘₯ -axis at the point ( 1 , 𝑏 ) . Find the value of π‘š + 2 π‘š 𝑏 .

Q11:

What are the coordinates of the vertex of the graph of 𝑓 ( π‘₯ ) = ( π‘₯ + 2 2 ) 2 ?

  • A ( 0 , βˆ’ 2 2 )
  • B ( 2 2 , 0 )
  • C ( 0 , 2 2 )
  • D ( βˆ’ 2 2 , 0 )

Q12:

For the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ + 3 2 , answer the following questions.

Find, by factoring, the zeros of the function.

  • A βˆ’ 3 , 1
  • B βˆ’ 1 , βˆ’ 3
  • C βˆ’ 4 , 1
  • D 1 , 3
  • E βˆ’ 1 , 4

Identify the graph of 𝑓 .

  • Athe red graph
  • Bthe blue graph
  • Cthe green graph

Write the equation for 𝑔 , the function that describes the blue graph.

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

Write the equation for β„Ž , the function that describes the green graph.

  • A β„Ž ( π‘₯ ) = βˆ’ ο€Ή π‘₯ βˆ’ 4 π‘₯ + 3  2
  • B β„Ž ( π‘₯ ) = π‘₯ + 4 π‘₯ βˆ’ 3 2
  • C β„Ž ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ + 3 2
  • D β„Ž ( π‘₯ ) = βˆ’ ο€Ή π‘₯ βˆ’ 4 π‘₯ βˆ’ 3  2
  • E β„Ž ( π‘₯ ) = π‘₯ + 4 π‘₯ + 3 2

Q13:

Find the coordinates of the vertex of the function 𝑓 ( π‘₯ ) = βˆ’ 7 π‘₯ + 7 π‘₯ + 5 2 .

  • A ο€Ό βˆ’ 1 2 , 5 1 4 
  • B ( 0 , 5 )
  • C ( 1 , 4 )
  • D ο€Ό 1 2 , 6 3 4 

Q14:

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

  • A ( 4 , 1 2 )
  • B ( 0 , βˆ’ 4 )
  • C ( βˆ’ 8 , βˆ’ 6 8 )
  • D ( βˆ’ 4 , βˆ’ 2 0 )

Q15:

Find the coordinates of the vertex of the function 𝑓 ( π‘₯ ) = π‘₯ + 4 π‘₯ βˆ’ 6 2 .

  • A ( 2 , βˆ’ 2 )
  • B ( 0 , βˆ’ 6 )
  • C ( βˆ’ 4 , βˆ’ 2 2 )
  • D ( βˆ’ 2 , βˆ’ 1 0 )

Q16:

Let 𝑓 be the function in the given table and 𝑔 ( π‘₯ ) = ( 2 π‘₯ + 1 ) βˆ’ 4 2 .

π‘₯ βˆ’4 βˆ’3 βˆ’2 βˆ’1 0 1 2 3
𝑓 ( π‘₯ ) 45 21 5 βˆ’3 βˆ’3 5 21 45

Which of the following is true?

  • AThey have the same zeros.
  • BThey are the same function.
  • CThey have the same vertex.
  • DThey have the same axis of symmetry.
  • EThey have the same sum of zeros.

Q17:

For the function 𝑓 ( π‘₯ ) = 3 0 π‘₯ + 9 π‘₯ βˆ’ 1 2 2 , answer the following questions.

Find, by factoring, the zeros of the function.

  • A 4 5 , βˆ’ 1 2
  • B βˆ’ 4 5 , βˆ’ 1 2
  • C βˆ’ 3 5 , 2 3
  • D βˆ’ 4 5 , 1 2
  • E 3 5 , 2 3

Identify the graph of 𝑓 .

  • Athe red graph
  • Bthe green graph
  • Cthe blue graph

Write the equation for 𝑔 , the function that describes the blue graph.

  • A 𝑔 ( π‘₯ ) = βˆ’ ο€Ή 3 0 π‘₯ + 9 π‘₯ + 1 2  2
  • B 𝑔 ( π‘₯ ) = βˆ’ ο€Ή 3 0 π‘₯ + 9 π‘₯ βˆ’ 1 2  2
  • C 𝑔 ( π‘₯ ) = 3 0 π‘₯ βˆ’ 9 π‘₯ βˆ’ 1 2 2
  • D 𝑔 ( π‘₯ ) = βˆ’ ο€Ή 3 0 π‘₯ βˆ’ 9 π‘₯ βˆ’ 1 2  2
  • E 𝑔 ( π‘₯ ) = 3 0 π‘₯ + 9 π‘₯ βˆ’ 1 2 2

Write the equation for β„Ž , the function that describes the green graph.

  • A β„Ž ( π‘₯ ) = βˆ’ ο€Ή 3 0 π‘₯ + 9 π‘₯ βˆ’ 1 2  2
  • B β„Ž ( π‘₯ ) = 3 0 π‘₯ βˆ’ 9 π‘₯ βˆ’ 1 2 2
  • C β„Ž ( π‘₯ ) = 3 0 π‘₯ + 9 π‘₯ βˆ’ 1 2 2
  • D β„Ž ( π‘₯ ) = βˆ’ ο€Ή 3 0 π‘₯ βˆ’ 9 π‘₯ βˆ’ 1 2  2
  • E β„Ž ( π‘₯ ) = βˆ’ ο€Ή 3 0 π‘₯ + 9 π‘₯ + 1 2  2

Q18:

The shown table is that of quadratic function 𝑓 .

π‘₯ 0 1 2 3 4 5
𝑓 ( π‘₯ ) βˆ’21 βˆ’5 3 3 βˆ’5 βˆ’21

Which of the following has an axis of symmetry closest to 𝑓 ?

  • A 𝑔 ( π‘₯ ) = 8 βˆ’ 5 ( 2 π‘₯ βˆ’ 7 ) 2
  • B 𝑔 ( π‘₯ ) = 7 βˆ’ | 1 βˆ’ π‘₯ |
  • C 𝑔 ( π‘₯ ) = ( π‘₯ + 1 ) ( 4 βˆ’ π‘₯ )
  • D 𝑔 ( π‘₯ ) = 7 π‘₯ + 4 βˆ’ 2 π‘₯ 2
  • E 𝑔 ( π‘₯ ) = 4 π‘₯ + 7 βˆ’ 2 π‘₯ 2

Q19:

Determine the quadratic function 𝑓 with the following properties:

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

Q20:

A stone is projected vertically upwards. Its height above the ground β„Ž after 𝑑 seconds is given by Find the maximum height the stone reaches.

Q21:

Determine the domain and the range of the function .

  • A The domain is , and the range is .
  • B The domain is , and the range is .
  • C The domain is , and the range is .
  • D The domain is , and the range is .
  • E The domain is , and the range is .

Q22:

Determine the domain and the range of the function .

  • A The domain is , and the range is .
  • B The domain is , and the range is .
  • C The domain is , and the range is .
  • D The domain is , and the range is .
  • E The domain is , and the range is .

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