Worksheet: Function Transformations: Translations

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In this worksheet, we will practice identifying function transformations involving horizontal and vertical shifts.

Q1:

The graph of function 𝑓 is produced from that of 𝑔(𝑥)=𝑥 by the following steps: a shift of 4 to the right, a dilation by a scale factor of 14, and a shift of 4 up. What is the function 𝑓?

  • A 𝑓 ( 𝑥 ) = 𝑥 4 − 2 𝑥 + 5
  • B 𝑓 ( 𝑥 ) = 𝑥 4 − 2 𝑥
  • C 𝑓 ( 𝑥 ) = 𝑥 4 − 2 𝑥 + 8
  • D 𝑓 ( 𝑥 ) = 𝑥 4 + 2 𝑥 + 8
  • E 𝑓 ( 𝑥 ) = 𝑥 4 + 2 𝑥 + 5

Q2:

The following graph is a transformation of the graph of 𝑦=|𝑥|. What is the function it represents? Write your answer in a form related to the transformation.

  • A 𝑦 = − 4 − | 𝑥 + 1 |
  • B 𝑦 = 4 | 𝑥 + 1 |
  • C 𝑦 = 4 − | 𝑥 + 1 |
  • D 𝑦 = 4 + | 𝑥 + 1 |
  • E 𝑦 = 4 − | 𝑥 − 1 |

Q3:

This is the graph of 𝑦=𝑔(𝑥).

Which of the following is the graph of 𝑔(−𝑥)?

  • A(c)
  • B(b)
  • C(a)

Q4:

Consider the following graph of the linear function 𝑓(𝑥).

Which of the following is the graph of the function 𝑓(2𝑥)?

  • A(d)
  • B(a)
  • C(b)
  • D(c)

Q5:

Which of the following is the graph of 𝑓(𝑥)=√𝑥−1?

  • A(c)
  • B(d)
  • C(a)
  • D(b)

Q6:

Which function is represented below?

  • A 𝑔 ( 𝑥 ) = √ 𝑥 − 3
  • B 𝑔 ( 𝑥 ) = − √ 𝑥 − 3
  • C 𝑔 ( 𝑥 ) = − √ 𝑥 + 3
  • D 𝑔 ( 𝑥 ) = − √ 𝑥 − 3
  • E 𝑔 ( 𝑥 ) = √ 𝑥 + 3

Q7:

The function 𝑦=(𝑥−1)(2𝑥−3)(4−𝑥) is translated 2 units in the direction of the positive x-axis. What is the equation of the resulting function?

  • A 𝑦 = ( 𝑥 − 3 ) ( 2 𝑥 − 7 ) ( 6 − 𝑥 )
  • B 𝑦 = ( 𝑥 − 1 ) ( 2 𝑥 − 3 ) ( 4 − 𝑥 ) + 2
  • C 𝑦 = ( 𝑥 + 1 ) ( 2 𝑥 + 1 ) ( 2 − 𝑥 )

Q8:

This is the graph of the exponential function 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=4−𝑓(2𝑥)?

  • A(a)
  • B(b)
  • C(c)

Q9:

Which of the following processes would you use to obtain the graph of 𝑦=−𝑓(−𝑥) from the graph of 𝑦=𝑓(𝑥)?

  1. reflect the graph in the 𝑥-axis
  2. reflect the graph in the 𝑦-axis
  3. reflect the graph in the line 𝑦=𝑥
  4. reflect the graph in the line 𝑦=−𝑥
  • Aa only
  • Bd only
  • Cb only
  • Dc and d
  • Ea and b

Q10:

The function 𝑦=𝑓(𝑥) is stretched in the horizontal direction by a scale factor of 13 and in the vertical direction by a scale factor of 13. Write, in terms of 𝑓(𝑥), the equation of the transformed function.

  • A 𝑦 = 1 3 𝑓 ( 3 𝑥 )
  • B 𝑦 = 1 3 𝑓 𝑥 3
  • C 𝑦 = 3 𝑓 𝑥 3
  • D 𝑦 = 𝑓 𝑥 + 1 3 + 1 3
  • E 𝑦 = 3 𝑓 ( 3 𝑥 )

Q11:

The function 𝑦=𝑓(𝑥) is translated eight down. Write, in terms of 𝑓(𝑥), the equation of the translated graph.

  • A 𝑦 = 𝑓 ( 𝑥 ) − 8
  • B 𝑦 = − 8 𝑓 ( 𝑥 )
  • C 𝑦 = 𝑓 ( 𝑥 + 8 )
  • D 𝑦 = 𝑓 ( 𝑥 − 8 )
  • E 𝑦 = 𝑓 ( − 8 𝑥 )

Q12:

The function 𝑦=𝑓(𝑥) is stretched in the horizontal direction by a scale factor of 2 and in the vertical direction by a scale factor of 2. Write, in terms of 𝑓(𝑥), the equation of the transformed function.

  • A 𝑦 = 1 2 𝑓 𝑥 2
  • B 𝑦 = 𝑓 𝑥 2
  • C 𝑦 = 2 𝑓 𝑥 2
  • D 𝑦 = 2 𝑓 ( 2 𝑥 )
  • E 𝑦 = 1 2 𝑓 ( 2 𝑥 )

Q13:

The function 𝑦=𝑓(𝑥) is stretched in the horizontal direction by a scale factor of 12. Write, in terms of 𝑓(𝑥), the equation of the transformed function.

  • A 𝑦 = 1 2 𝑓 ( 𝑥 )
  • B 𝑦 = 𝑓 ( 𝑥 2 )
  • C 𝑦 = 𝑓 ( 𝑥 + 2 )
  • D 𝑦 = 2 𝑓 ( 𝑥 )
  • E 𝑦 = 𝑓 ( 2 𝑥 )

Q14:

Daniel thinks that he can map the graph of the function 𝑓(𝑥)=2𝑥−1 to the graph of any other linear function by a translation followed by a stretch. This is equivalent to saying that any linear function 𝑔(𝑥) can be written in the form 𝐴𝑓(𝑥−𝑏) for suitable values of 𝐴 and 𝑏.

Is he right?

  • Ano
  • Byes

Suppose that 𝑔(𝑥)=𝑚𝑥+𝑐 and 𝑓(𝑥) is as before. Find 𝐴 and 𝑏 in terms of 𝑚 and 𝑐 when it is possible to write 𝑔(𝑥) in the form 𝐴𝑓(𝑥−𝑏).

  • A 𝐴 = 3 𝑚 4 , 𝑏 = 1 2 1 − 2 𝑐 𝑚
  • B 𝐴 = 𝑚 2 , 𝑏 = 1 2 1 − 2 𝑐 𝑚
  • C 𝐴 = 𝑚 2 , 𝑏 = 1 2 1 + 2 𝑐 𝑚
  • D 𝐴 = 1 2 1 + 2 𝑐 𝑚 , 𝑏 = 𝑐 𝑚 − 2
  • E 𝐴 = 2 𝑚 , 𝑏 = 1 − 2 𝑐 𝑚

Let 𝑓(𝑥)=43𝑥+4 and 𝑔(𝑥)=2𝑥−2. Find the values of 𝐴 and 𝑏 for which 𝑔(𝑥)=𝐴𝑓(𝑥−𝑏). Use the following graph to help you.

  • A 𝐴 = 3 4 , 𝑏 = 3
  • B 𝐴 = 3 2 , 𝑏 = − 9 4
  • C 𝐴 = 2 3 , 𝑏 = 1 1 4
  • D 𝐴 = 3 2 , 𝑏 = 4
  • E 𝐴 = 4 3 , 𝑏 = 4

Q15:

Graphs 𝐴 and 𝐵 in the diagram are the graphs of square root functions. They are symmetric about the origin. The equation of graph 𝐴 is 𝑦=13√𝑥+2+1. Knowing that a point reflection about the origin is equivalent to a reflection in the 𝑥-axis followed by a reflection in the 𝑦-axis, find the equation of graph 𝐵.

  • A 𝑦 = − 1 3 √ − 𝑥 + 2 − 1
  • B 𝑦 = 1 3 √ − 𝑥 + 2 − 1
  • C 𝑦 = − 1 3 √ 𝑥 − 2 − 1
  • D 𝑦 = − 1 3 √ − 𝑥 − 2 − 1
  • E 𝑦 = − 1 3 √ 𝑥 + 2 − 1

Q16:

Consider the function 𝑓 given by 𝑓(𝑥)=√𝑥+1−2.

Which of the graphs in the given diagram is the reflection of the graph of 𝑓 in the 𝑥-axis?

  • AH
  • BE
  • CF
  • DG

Write its equation.

  • A 𝑦 = √ − 𝑥 + 1 − 2
  • B 𝑦 = − √ 𝑥 + 1 − 2
  • C 𝑦 = − √ 𝑥 + 1 + 2
  • D 𝑦 = √ − 𝑥 + 1 + 2

Q17:

Consider the function 𝑓 given by 𝑓(𝑥)=√𝑥+1+2.

Which of the graphs in the given diagram is the reflection of the graph of 𝑓 in the 𝑦-axis?

  • AG
  • BC
  • CE
  • DA

Write its equation.

  • A 𝑦 = √ 𝑥 + 1 + 2
  • B 𝑦 = − √ 𝑥 + 1 + 2
  • C 𝑦 = √ − 𝑥 + 1 + 2
  • D 𝑦 = − √ − 𝑥 + 1 + 2

Q18:

Consider the function 𝑓(𝑥)=√𝑥−1+2.

Which of the following is the graph of 𝑦=𝑓(𝑥)?

  • AC
  • BB
  • CA
  • DD

State the domain and range of 𝑓(𝑥).

  • Adomain: 𝑥≥−1, range: 𝑦≥2
  • Bdomain: 𝑥≥1, range: 𝑦≥2
  • Cdomain: 𝑥≥1, range: 𝑦≥−2
  • Ddomain: 𝑥≥−1, range: 𝑦≥−2

Q19:

Consider the function 𝑓(𝑥)=√𝑥+1+2.

Which of the graphs shown in the given diagram is the reflection of 𝑓(𝑥) in the 𝑥-axis?