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Lesson: Features of Quadratic Functions

In this lesson, we will learn how to identify features of quadratic equations, such as its vertex, maximum or minimum value, axis of symmetry, domain, and range.

Sample Question Videos

01:33

Worksheet • 22 Questions • 1 Video

Q1:

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

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

Q2:

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

Q3:

Find the axis of symmetry of the graph of 𝑓 ( π‘₯ ) = 4 π‘₯ + 4 π‘₯ βˆ’ 3 2 .

  • A π‘₯ = βˆ’ 1 2
  • B π‘₯ = βˆ’ 3 4
  • C π‘₯ = 1 2
  • D π‘₯ = 4
  • E π‘₯ = βˆ’ 4

Q4:

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?

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 = βˆ’ 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 = 1 6 , 𝑧 = 0

Q6:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = 4 ( π‘₯ βˆ’ 4 ) βˆ’ 3 2 .

  • A The domain is ℝ , and the range is [ βˆ’ 3 , ∞ [ .
  • B The domain is ℝ , and the range is ] βˆ’ 3 , ∞ [ .
  • C The domain is [ βˆ’ 3 , ∞ [ , and the range is ℝ .
  • D The domain is ℝ βˆ’ { βˆ’ 3 } , and the range is ℝ βˆ’ { 4 } .
  • E The domain is ℝ βˆ’ { 4 } , and the range is ℝ βˆ’ { βˆ’ 3 } .

Q7:

Determine the domain and the range of the function 𝑓 ( π‘₯ ) = π‘₯ + 8 π‘₯ + 2 0 2 .

  • A The domain is ℝ , and the range is [ 4 , ∞ [ .
  • B The domain is ℝ βˆ’ { 4 } , and the range is ℝ βˆ’ { βˆ’ 4 } .
  • C The domain is [ 4 , ∞ [ , and the range is ℝ .
  • D The domain is ℝ βˆ’ { βˆ’ 4 } , and the range is ℝ βˆ’ { 4 } .
  • E The domain is ℝ , and the range is ] 4 , ∞ [ .

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

Identify the graph of 𝑓 .

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

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

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

Q9:

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

Find, by factoring, the zeros of the function.

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

Identify the graph of 𝑓 .

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

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

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

Q10:

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

Identify the graph of 𝑓 .

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

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

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

Q11:

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

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

Q12:

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?

Q13:

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

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

Q14:

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

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

Q15:

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

Q16:

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 The function does not exist.
  • B 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 3 ) βˆ’ 1 7 2
  • C 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ βˆ’ 3 ) βˆ’ 1 7 2
  • D 𝑓 ( π‘₯ ) = 2 2 ( π‘₯ βˆ’ 3 ) + 1 7 2
  • E 𝑓 ( π‘₯ ) = ( π‘₯ + 3 ) βˆ’ 1 7 2

Q17:

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.

Q18:

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 .

Q19:

An object’s height in feet, 𝑦 , is the function 𝑦 = βˆ’ π‘₯ + 9 6 π‘₯  of the horizontal distance traveled, π‘₯ feet, from where it is projected. What is the maximum height of this motion?

Q20:

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

Q21:

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

Q22:

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

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