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12,634
  • C−12,514
  • 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.

  • A𝑥=−4
  • B𝑥=4
  • C𝑥=−34
  • D𝑥=−12
  • E𝑥=12

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.

  • A𝑟=−8, 𝑡=8, 𝑧=0
  • B𝑟=8, 𝑡=−16, 𝑧=0
  • C𝑟=16, 𝑡=8, 𝑧=0
  • D𝑟=8, 𝑡=16, 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−74,3
  • C−3,7
  • D−3,−74
  • E−3,74

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.

  • A𝑔(𝑥)=−−4𝑥+5𝑥+21
  • B𝑔(𝑥)=−4𝑥−5𝑥+21
  • C𝑔(𝑥)=−4𝑥+5𝑥+21
  • D𝑔(𝑥)=−−4𝑥−5𝑥+21
  • E𝑔(𝑥)=−−4𝑥−5𝑥−21

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

  • Aℎ(𝑥)=−4𝑥+5𝑥−21
  • Bℎ(𝑥)=−−4𝑥−5𝑥+21
  • Cℎ(𝑥)=−−4𝑥+5𝑥+21
  • Dℎ(𝑥)=−4𝑥+5𝑥+21
  • Eℎ(𝑥)=−4𝑥−5𝑥+21

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
  • E1,3

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.

  • A𝑔(𝑥)=𝑥+4𝑥−3
  • B𝑔(𝑥)=𝑥+4𝑥+3
  • C𝑔(𝑥)=−𝑥+4𝑥−3
  • D𝑔(𝑥)=−𝑥+4𝑥+3
  • E𝑔(𝑥)=−𝑥−4𝑥+3

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

  • Aℎ(𝑥)=𝑥−4𝑥+3
  • Bℎ(𝑥)=𝑥+4𝑥−3
  • Cℎ(𝑥)=−𝑥−4𝑥−3
  • Dℎ(𝑥)=𝑥+4𝑥+3
  • Eℎ(𝑥)=−𝑥−4𝑥+3

Q10:

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

Find, by factoring, the zeros of the function.

  • A−35,23
  • B−45,12
  • C35,23
  • D45,−12
  • E−45,−12

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.

  • A𝑔(𝑥)=30𝑥+9𝑥−12
  • B𝑔(𝑥)=−30𝑥+9𝑥−12
  • C𝑔(𝑥)=−30𝑥−9𝑥−12
  • D𝑔(𝑥)=−30𝑥+9𝑥+12
  • E𝑔(𝑥)=30𝑥−9𝑥−12

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

  • Aℎ(𝑥)=−30𝑥−9𝑥−12
  • Bℎ(𝑥)=30𝑥−9𝑥−12
  • Cℎ(𝑥)=−30𝑥+9𝑥+12
  • Dℎ(𝑥)=−30𝑥+9𝑥−12
  • Eℎ(𝑥)=30𝑥+9𝑥−12

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,−22)
  • B(22,0)
  • C(−22,0)
  • D(0,22)

Q14:

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

𝑥−4−3−2−10123
𝑓(𝑥)45215−3−352145

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.

𝑥012345
𝑓(𝑥)−21−533−5−21

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

  • A𝑔(𝑥)=4𝑥+7−2𝑥
  • B𝑔(𝑥)=8−5(2𝑥−7)
  • C𝑔(𝑥)=(𝑥+1)(4−𝑥)
  • D𝑔(𝑥)=7𝑥+4−2𝑥
  • E𝑔(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→−∞.
  • A𝑓(𝑥)=(𝑥+3)−17
  • BThe function does not exist.
  • C𝑓(𝑥)=22(𝑥−3)−17
  • D𝑓(𝑥)=22(𝑥−3)+17
  • E𝑓(𝑥)=(𝑥−3)−17

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−12,−714
  • C12,−434
  • 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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