Worksheet: Features of Quadratic Functions

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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 function 𝑓 ( 𝑥 ) = − 7 𝑥 + 7 𝑥 + 5 .

  • A − 1 2 , 5 1 4
  • B ( 0 , 5 )
  • C ( 1 , 4 )
  • D 1 2 , 6 3 4

Q2:

Find the coordinates of the vertex of the graph of 𝑓 ( 𝑥 ) = 𝑥 − 6 𝑥 − 4 . 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 , 3 ) , the value of the function at the vertex is − 1 3 , it is a minimum value.
  • B The vertex is ( 3 , − 1 3 ) , the value of the function at the vertex is 3, it is a minimum value.
  • C The vertex is ( − 1 3 , 3 ) , the value of the function at the vertex is 3, it is a maximum value.
  • D The vertex is ( 3 , − 1 3 ) , the value of the function at the vertex is − 1 3 , it is a minimum value.
  • E The vertex is ( 3 , − 1 3 ) , the value of the function at the vertex is − 1 3 , it is a maximum value.

Q3:

Find the axis of symmetry of the graph of 𝑓 ( 𝑥 ) = 4 𝑥 + 4 𝑥 − 3 2 .

  • A 𝑥 = − 4
  • B 𝑥 = 1 2
  • C 𝑥 = 4
  • D 𝑥 = − 1 2
  • E 𝑥 = − 3 4

Q4:

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

Q5:

The graph of the function 𝑓 ( 𝑥 ) = 𝑟 𝑥 + 𝑡 𝑥 + 𝑧 passes through the point ( 0 , 0 ) . Given that the minimum value of the function is − 8 , and the axis of symmetry is the line 𝑥 = 1 , find the the values of 𝑟 , 𝑡 , and 𝑧 .

  • A 𝑟 = 8 , 𝑡 = 1 6 , 𝑧 = 0
  • B 𝑟 = − 8 , 𝑡 = 8 , 𝑧 = 0
  • C 𝑟 = 1 6 , 𝑡 = 8 , 𝑧 = 0
  • D 𝑟 = 8 , 𝑡 = − 1 6 , 𝑧 = 0

Q6:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 4 ( 𝑥 − 4 ) − 3 .

  • A The domain is ℝ − { 4 } , and the range is ℝ − { − 3 } .
  • B The domain is [ − 3 , ∞ ) , and the range is ℝ .
  • C The domain is ℝ − { − 3 } , and the range is ℝ − { 4 } .
  • D The domain is ℝ , and the range is [ − 3 , ∞ ) .
  • E The domain is ℝ , and the range is ( − 3 , ∞ ) .

Q7:

Determine the domain and the range of the function 𝑓 ( 𝑥 ) = 𝑥 + 8 𝑥 + 2 0 .

  • A The domain is ℝ , and the range is ( 4 , ∞ ) .
  • B The domain is [ 4 , ∞ ) , and the range is ℝ .
  • C The domain is ℝ − { − 4 } , and the range is ℝ − { 4 } .
  • D The domain is ℝ , and the range is [ 4 , ∞ ) .
  • E The domain is ℝ − { 4 } , and the range is ℝ − { − 4 } .

Q8:

For the function 𝑓 ( 𝑥 ) = − 4 𝑥 + 5 𝑥 + 2 1 2 , answer the following questions.

Find, by factoring, the zeros of the function.

  • A − 3 , 7 4
  • B − 3 , − 7 4
  • C − 7 , 3
  • D − 7 4 , 3
  • E − 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 ( 𝑥 ) = − − 4 𝑥 − 5 𝑥 − 2 1 2
  • B 𝑔 ( 𝑥 ) = − 4 𝑥 + 5 𝑥 + 2 1 2
  • C 𝑔 ( 𝑥 ) = − − 4 𝑥 + 5 𝑥 + 2 1 2
  • D 𝑔 ( 𝑥 ) = − 4 𝑥 − 5 𝑥 + 2 1 2
  • E 𝑔 ( 𝑥 ) = − − 4 𝑥 − 5 𝑥 + 2 1 2

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

  • A ℎ ( 𝑥 ) = − 4 𝑥 − 5 𝑥 + 2 1 2
  • B ℎ ( 𝑥 ) = − 4 𝑥 + 5 𝑥 − 2 1 2
  • C ℎ ( 𝑥 ) = − 4 𝑥 + 5 𝑥 + 2 1 2
  • D ℎ ( 𝑥 ) = − − 4 𝑥 − 5 𝑥 + 2 1 2
  • E ℎ ( 𝑥 ) = − − 4 𝑥 + 5 𝑥 + 2 1 2

Q9:

For the function 𝑓 ( 𝑥 ) = 𝑥 − 4 𝑥 + 3 2 , answer the following questions.

Find, by factoring, the zeros of the function.

  • A − 3 , 1
  • B − 1 , − 3
  • C − 4 , 1
  • D 1 , 3
  • E − 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 ( 𝑥 ) = 𝑥 + 4 𝑥 − 3 2
  • B 𝑔 ( 𝑥 ) = − 𝑥 + 4 𝑥 − 3 2
  • C 𝑔 ( 𝑥 ) = 𝑥 + 4 𝑥 + 3 2
  • D 𝑔 ( 𝑥 ) = − 𝑥 + 4 𝑥 + 3 2
  • E 𝑔 ( 𝑥 ) = − 𝑥 − 4 𝑥 + 3 2

Write the equation for ℎ , the function that describes the green graph.

  • A ℎ ( 𝑥 ) = − 𝑥 − 4 𝑥 + 3 2
  • B ℎ ( 𝑥 ) = 𝑥 + 4 𝑥 − 3 2
  • C ℎ ( 𝑥 ) = 𝑥 − 4 𝑥 + 3 2
  • D ℎ ( 𝑥 ) = − 𝑥 − 4 𝑥 − 3 2
  • E ℎ ( 𝑥 ) = 𝑥 + 4 𝑥 + 3 2

Q10:

For the function 𝑓 ( 𝑥 ) = 3 0 𝑥 + 9 𝑥 − 1 2 2 , answer the following questions.

Find, by factoring, the zeros of the function.

  • A 4 5 , − 1 2
  • B − 4 5 , − 1 2
  • C − 3 5 , 2 3
  • D − 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 ( 𝑥 ) = − 3 0 𝑥 + 9 𝑥 + 1 2 2
  • B 𝑔 ( 𝑥 ) = − 3 0 𝑥 + 9 𝑥 − 1 2 2
  • C 𝑔 ( 𝑥 ) = 3 0 𝑥 − 9 𝑥 − 1 2 2
  • D 𝑔 ( 𝑥 ) = − 3 0 𝑥 − 9 𝑥 − 1 2 2
  • E 𝑔 ( 𝑥 ) = 3 0 𝑥 + 9 𝑥 − 1 2 2

Write the equation for ℎ , the function that describes the green graph.

  • A ℎ ( 𝑥 ) = − 3 0 𝑥 + 9 𝑥 − 1 2 2
  • B ℎ ( 𝑥 ) = 3 0 𝑥 − 9 𝑥 − 1 2 2
  • C ℎ ( 𝑥 ) = 3 0 𝑥 + 9 𝑥 − 1 2 2
  • D ℎ ( 𝑥 ) = − 3 0 𝑥 − 9 𝑥 − 1 2 2
  • E ℎ ( 𝑥 ) = − 3 0 𝑥 + 9 𝑥 + 1 2 2

Q11:

The following statements refer to the function 𝑓 ( 𝑥 ) = 𝑎 𝑥 + 𝑏 𝑥 + 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 𝑥 = − 𝑏 2 𝑎 , 𝑦 = 𝑓 − 𝑏 2 𝑎 .

Q12:

A ball is hurled upward from the top of a building. Its height in feet at time 𝑡 , in seconds, can be described by the function 𝐻 ( 𝑡 ) = − 1 6 𝑡 + 6 4 𝑡 + 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 ( 𝑥 ) = ( 𝑥 + 2 2 ) ?

  • A ( 0 , − 2 2 )
  • B ( 2 2 , 0 )
  • C ( 0 , 2 2 )
  • D ( − 2 2 , 0 )

Q14:

Let 𝑓 be the function in the given table and 𝑔 ( 𝑥 ) = ( 2 𝑥 + 1 ) − 4 .

𝑥 −4 −3 −2 −1 0 1 2 3
𝑓 ( 𝑥 ) 45 21 5 −3 −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.

Q15:

The shown table is that of quadratic function 𝑓 .

𝑥 0 1 2 3 4 5
𝑓 ( 𝑥 ) −21 −5 3 3 −5 −21

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

  • A 𝑔 ( 𝑥 ) = 8 − 5 ( 2 𝑥 − 7 )
  • B 𝑔 ( 𝑥 ) = 7 − | 1 − 𝑥 |
  • C 𝑔 ( 𝑥 ) = ( 𝑥 + 1 ) ( 4 − 𝑥 )
  • D 𝑔 ( 𝑥 ) = 7 𝑥 + 4 − 2 𝑥
  • E 𝑔 ( 𝑥 ) = 4 𝑥 + 7 − 2 𝑥

Q16:

Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at ( 3 , − 1 7 )
  • 𝑓 ( 4 ) = 5
  • 𝑓 ( 𝑥 ) → − ∞ as 𝑥 → − ∞ .
  • A 𝑓 ( 𝑥 ) = ( 𝑥 + 3 ) − 1 7 2
  • B 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 − 3 ) − 1 7 2
  • C 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 − 3 ) + 1 7 2
  • D The function does not exist.
  • E 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) − 1 7 2

Q17:

A stone is projected vertically upwards. Its height above the ground ℎ after 𝑡 seconds is given by ℎ = 4 . 9 𝑡 − 4 . 9 𝑡 𝑡 ≥ 0 . m , Find the maximum height the stone reaches.

Q18:

The function 𝑓 ( 𝑥 ) = 𝑚 − 3 𝑥 intersects the 𝑥 -axis at the point ( 1 , 𝑏 ) . Find the value of 𝑚 + 2 𝑚 .

Q19:

An object’s height in feet, 𝑦 , is the function 𝑦 = − 𝑥 + 9 6 𝑥 of the horizontal distance traveled, 𝑥 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 ( 𝑥 ) = 𝑥 + 4 𝑥 + 5 . State the value of the function at the vertex and determine whether it is a maximum or minimum value.

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

Q21:

Find the coordinates of the vertex of the graph of 𝑓 ( 𝑥 ) = − 𝑥 + 2 𝑥 + 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.

Q22:

Find the coordinates of the vertex of the function 𝑓 ( 𝑥 ) = 9 𝑥 + 9 𝑥 − 5 .

  • A 1 2 , − 4 3 4
  • B ( 0 , − 5 )
  • C ( − 1 , − 6 )
  • D − 1 2 , − 7 1 4

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