Worksheet: Features of Quadratic Functions

In this worksheet, we will practice identifying features of quadratic functions, such as its vertex, extrema, axis of symmetry, domain, and range.

Q1:

Find the coordinates of the vertex of the function 𝑓(𝑥)=−7𝑥+7𝑥+5.

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

Q2:

Find the coordinates of the vertex of the graph of 𝑓(𝑥)=𝑥−6𝑥−4. State the value of the function at the vertex and determine whether it is a maximum or minimum value.

  • AThe vertex is (−13,3), the value of the function at the vertex is 3, it is a maximum value.
  • BThe vertex is (3,−13), the value of the function at the vertex is −13, it is a minimum value.
  • CThe vertex is (3,−13), the value of the function at the vertex is 3, it is a minimum value.
  • DThe vertex is (−13,3), the value of the function at the vertex is −13, it is a minimum value.
  • EThe vertex is (3,−13), the value of the function at the vertex is −13, it is a maximum value.

Q3:

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

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

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)=𝑟𝑥+𝑡𝑥+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 = 8 , 𝑧 = 0
  • B 𝑟 = 8 , 𝑡 = − 1 6 , 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.

  • AThe domain is [−3,∞), and the range is ℝ.
  • BThe domain is ℝ−{−3}, and the range is ℝ−{4}.
  • CThe domain is ℝ, and the range is [−3,∞).
  • DThe domain is ℝ, and the range is (−3,∞).
  • EThe domain is ℝ−{4}, and the range is ℝ−{−3}.

Q7:

Determine the domain and the range of the function 𝑓(𝑥)=𝑥+8𝑥+20.

  • AThe domain is ℝ, and the range is (4,∞).
  • BThe domain is ℝ−{−4}, and the range is ℝ−{4}.
  • CThe domain is [4,∞), and the range is ℝ.
  • DThe domain is ℝ, and the range is [4,∞).
  • EThe domain is ℝ−{4}, and the range is ℝ−{−4}.

Q8:

For the function 𝑓(𝑥)=−4𝑥+5𝑥+21, answer the following questions.

Find, by factoring, the zeros of the function.

  • A − 7 , 3
  • B − 7 4 , 3
  • C − 3 , 7
  • D − 3 , − 7 4
  • E − 3 , 7 4

Identify the graph of 𝑓.

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

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

  • A 𝑔 ( 𝑥 ) = −  − 4 𝑥 + 5 𝑥 + 2 1  
  • B 𝑔 ( 𝑥 ) = − 4 𝑥 − 5 𝑥 + 2 1 
  • C 𝑔 ( 𝑥 ) = − 4 𝑥 + 5 𝑥 + 2 1 
  • D 𝑔 ( 𝑥 ) = −  − 4 𝑥 − 5 𝑥 + 2 1  
  • E 𝑔 ( 𝑥 ) = −  − 4 𝑥 − 5 𝑥 − 2 1  

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

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

Q9:

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

Find, by factoring, the zeros of the function.

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

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

  • A ℎ ( 𝑥 ) = 𝑥 − 4 𝑥 + 3 
  • B ℎ ( 𝑥 ) = 𝑥 + 4 𝑥 − 3 
  • C ℎ ( 𝑥 ) = −  𝑥 − 4 𝑥 − 3  
  • D ℎ ( 𝑥 ) = 𝑥 + 4 𝑥 + 3 
  • E ℎ ( 𝑥 ) = −  𝑥 − 4 𝑥 + 3  

Q10:

For the function 𝑓(𝑥)=30𝑥+9𝑥−12, answer the following questions.

Find, by factoring, the zeros of the function.

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

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

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

Q11:

The following statements refer to the function 𝑓(𝑥)=𝑎𝑥+𝑏𝑥+𝑐. Which statement is true?

  • AThe function has a vertex at 𝑥=−𝑏2𝑎,𝑦=𝑓−𝑏2𝑎.
  • BThe function has one relative maximum.
  • CThe function has two roots.
  • 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 𝑡, in seconds, can be described by the function 𝐻(𝑡)=−16𝑡+64𝑡+120. 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+22)?

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

Q14:

Let 𝑓 be the function in the given table and 𝑔(𝑥)=(2𝑥+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 axis of symmetry.
  • BThey have the same sum of zeros.
  • CThey have the same zeros.
  • DThey are the same function.
  • EThey have the same vertex.

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

Q16:

Determine the quadratic function 𝑓 with the following properties:

  • its graph has a vertex at (3,−17)
  • 𝑓 ( 4 ) = 5
  • 𝑓 ( 𝑥 ) → − ∞ as 𝑥→−∞.
  • A 𝑓 ( 𝑥 ) = ( 𝑥 + 3 ) − 1 7 
  • BThe function does not exist.
  • C 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 − 3 ) − 1 7 
  • D 𝑓 ( 𝑥 ) = 2 2 ( 𝑥 − 3 ) + 1 7 
  • E 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) − 1 7 

Q17:

A stone is projected vertically upward. Its height above the ground, ℎ, after 𝑡 seconds is given by ℎ=4.9𝑡−4.9𝑡𝑡≥0m,. Find the maximum height the stone reaches.

Q18:

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

Q19:

An object’s height in feet, 𝑦, is the function 𝑦=−𝑥+96𝑥 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+4𝑥+5. State the value of the function at the vertex and determine whether it is a maximum or minimum value.

  • AThe vertex is (1,−2), the value of the function at the vertex is −2, it is a maximum value.
  • BThe vertex is (−2,1), the value of the function at the vertex is 1, it is a minimum value.
  • CThe vertex is (−2,1), the value of the function at the vertex is −2, it is a minimum value.
  • DThe vertex is (1,−2), the value of the function at the vertex is 1, it is a minimum value.
  • EThe 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+2𝑥+2. State the value of the function at the vertex and determine whether it is a maximum or minimum value.

  • AThe vertex is (3,1), the value of the function at the vertex is 1, it is a minimum value.
  • BThe vertex is (1,3), the value of the function at the vertex is 3, it is a maximum value.
  • CThe vertex is (1,3), the value of the function at the vertex is 1, it is a maximum value.
  • DThe vertex is (3,1), the value of the function at the vertex is 3, it is a maximum value.
  • EThe 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)=9𝑥+9𝑥−5.

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

Q23:

Let 𝑓(𝑥)=−2𝑥+4𝑥+6 and 𝑦=𝑔(𝑥) be the function whose graph is shown.

Which of the following statements is true?

  • AThe minimum value of 𝑓 is greater than the maximum value of 𝑔.
  • BThe sums of zeros of the two functions are the same.
  • COn the interval −2≤𝑥≤0, function 𝑓 has the higher average rate of change.
  • DOn the interval 0<𝑥<3, function 𝑓 has the lower average rate of change.
  • EBoth functions have the same axis of symmetry.

Q24:

A quadratic function 𝑓 has distinct positive roots. Which of the following must be true about its graph?

  • AThe parabola must be concave up.
  • BThe axis of symmetry must be 𝑥+𝑐=0 for a positive value 𝑐.
  • CThe axis of symmetry must be 𝑥=𝑐 for a positive value 𝑐.
  • DIts vertex must be above the 𝑥-axis.
  • EThe parabola must be concave down.

Q25:

Is any quadratic function a nonlinear function?

  • Ano
  • Byes

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