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Worksheet: Transformations of Functions

In this worksheet, we will practice finding the transformation of a given function.

Q1:

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

Q2:

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 1 4 , and a shift of 4 up. What is the function 𝑓 ?

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

Q3:

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 |

Q4:

The function 𝑦 = 𝑓 ( 𝑥 ) is stretched in the horizontal direction by a scale factor of 1 2 . Write, in terms of 𝑓 ( 𝑥 ) , the equation of the transformed function.

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

Q5:

The figure shows the graph of 𝑦 = 𝑓 ( 𝑥 ) and the point 𝐴 . The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 = 𝑓 ( 𝑥 ) − 2 .

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

Q6:

Daniel thinks that he can map the graph of the function 𝑓 ( 𝑥 ) = 2 𝑥 − 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 𝐴 𝑓 ( 𝑥 − 𝑏 ) for suitable values of 𝐴 and 𝑏 .

Is he right?

  • Ano
  • Byes

Suppose that 𝑔 ( 𝑥 ) = 𝑚 𝑥 + 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 𝐴 𝑓 ( 𝑥 − 𝑏 ) .

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

Let 𝑓 ( 𝑥 ) = 4 3 𝑥 + 4 and 𝑔 ( 𝑥 ) = 2 𝑥 − 2 . Find the values of 𝐴 and 𝑏 for which 𝑔 ( 𝑥 ) = 𝐴 𝑓 ( 𝑥 − 𝑏 ) . Use the following graph to help you.

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

Q7:

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 = 1 3 √ 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

Q8:

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
  • BF
  • CE
  • DG

Write its equation.

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

Q9:

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
  • BE
  • CC
  • DA

Write its equation.

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

Q10:

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

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

  • AC
  • BA
  • CD
  • DB

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

Q11:

The figure shows the graph of 𝑦 = 𝑓 ( 𝑥 ) and the point 𝐴 . The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 = 𝑓 ( 𝑥 − 1 ) + 4 .

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

Q12:

The red graph in the figure has equation 𝑦 = 𝑓 ( 𝑥 ) and the black graph has equation 𝑦 = 𝑔 ( 𝑥 ) . Express 𝑔 ( 𝑥 ) as a transformation of 𝑓 ( 𝑥 ) .

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

Q13:

This is the graph of 𝑦 = 𝑔 ( 𝑥 ) .

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

  • A(c)
  • B(a)
  • C(b)

Q14:

The figure shows the graph of 𝑦 = 𝑓 ( 𝑥 ) and point 𝐴 . Point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 = 𝑓 ( 𝑥 + 4 ) .

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

Q15:

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?

  • AF
  • BG
  • CE
  • DH

Write its equation.

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

Q16:

The figure shows the graph of 𝑦 = 𝑓 ( 𝑥 ) and point 𝐴 , which is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 = 𝑓 ( 𝑥 − 3 ) .