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Worksheet: Transformations of Functions

Q1:

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
  • Ab only
  • Ba only
  • Cc and d
  • Da and b
  • Ed only

Q2:

This is the graph of 𝑦 = 𝑔 ( 𝑥 ) .

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

  • A(c)
  • B(a)
  • C(b)

Q3:

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

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

  • A(c)
  • B(a)
  • C(d)
  • D(b)

Q4:

The function 𝑦 = 𝑓 ( 𝑥 ) is stretched in the horizontal direction by a scale factor of 1 3 and in the vertical direction by a scale factor of 1 3 . Write, in terms of 𝑓 ( 𝑥 ) , the equation of the transformed function.

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

Q5:

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 = 𝑓 ( 𝑥 8 )
  • C 𝑦 = 8 𝑓 ( 𝑥 )
  • D 𝑦 = 𝑓 ( 𝑥 ) 8
  • E 𝑦 = 𝑓 ( 8 𝑥 )

Q6:

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

Q7:

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 )

Q8:

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

Q9:

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

Q10:

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

Q11:

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

Q12:

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

Q13:

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

Q14:

Consider the function 𝑓 ( 𝑥 ) = 3 𝑥 .

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

  • AC
  • BB
  • CD
  • DA

How would you describe the transformation from 𝑔 ( 𝑥 ) = 𝑥 to 𝑓 ( 𝑥 ) ?

  • Adilation in the vertical direction by a factor of 3
  • Bdilation in the horizontal direction by a factor of 3
  • Ctranslation three units up

State the domain and range of 𝑓 ( 𝑥 ) .

  • A domain: 𝑥 0 , range: 𝑦 3
  • Bdomain: 𝑥 3 , range: 𝑦 0
  • Cdomain: 𝑥 0 , range: 𝑦 0

Q15:

Consider the function 𝑓 given by 𝑓 ( 𝑥 ) = 𝑥 .

The function 𝑔 ( 𝑥 ) is obtained by dilating 𝑓 in the horizontal direction by a factor 𝑏 and translating it 𝑐 units horizontally. Write the equation for 𝑔 ( 𝑥 ) .

  • A 𝑔 ( 𝑥 ) = 𝑏 𝑥 𝑐
  • B 𝑔 ( 𝑥 ) = 𝑐 𝑥 𝑏
  • C 𝑔 ( 𝑥 ) = 𝑏 𝑥 𝑐 𝑏
  • D 𝑔 ( 𝑥 ) = 𝑏 𝑥 𝑐

Find 𝑥 0 so that 𝑔 ( 𝑥 ) = 0 0 .

  • A 𝑥 = 𝑏 𝑐 0
  • B 𝑥 = 𝑐 𝑏 0
  • C 𝑥 = 𝑏 𝑐 0
  • D 𝑥 = 𝑐 𝑏 0

Find 𝑥 1 so that 𝑔 ( 𝑥 ) = 1 1 .

  • A 𝑥 = 𝑐 + 1 𝑏 1
  • B 𝑥 = 1 𝑐 𝑏 1
  • C 𝑥 = 𝑏 + 1 𝑐 1
  • D 𝑥 = 1 𝑏 𝑐 1

Use your previous answers to find the equation of the graph shown in the given diagram.

  • A 𝑦 = 2 𝑥 1
  • B 𝑦 = 2 𝑥 1 2
  • C 𝑦 = 2 𝑥 1 2
  • D 𝑦 = 𝑥 1 2

Q16:

Consider the root function 𝑓 ( 𝑥 ) = 𝑥 .

The function 𝑔 ( 𝑥 ) has been obtained by translating 𝑓 ( 𝑥 ) three units down and five units to the left. Write its equation.

  • A 𝑔 ( 𝑥 ) = 𝑥 5 3
  • B 𝑔 ( 𝑥 ) = 𝑥 + 5 + 3
  • C 𝑔 ( 𝑥 ) = 𝑥 5 + 3
  • D 𝑔 ( 𝑥 ) = 𝑥 + 5 3
  • E 𝑔 ( 𝑥 ) = 𝑥 + 3 5

State the domain and range of 𝑔 ( 𝑥 ) .

  • Adomain: 𝑥 5 , range: 𝑦 3
  • Bdomain: 𝑥 5 , range: 𝑦 3
  • Cdomain: 𝑥 5 , range: 𝑦 3
  • Ddomain: 𝑥 5 , range: 𝑦 3
  • Edomain: 𝑥 3 , range: 𝑦 5

Q17:

Consider the function 𝑓 given by 𝑓 ( 𝑥 ) = 𝑥 .

The function 𝑔 is obtained by dilating 𝑓 in the vertical direction by a factor 𝑐 and translating it units horizontally and 𝑘 units vertically. Write an equation for 𝑔 ( 𝑥 ) .

  • A 𝑔 ( 𝑥 ) = 𝑐 𝑥 𝑘 +
  • B 𝑔 ( 𝑥 ) = 𝑐 𝑥 + 𝑘
  • C 𝑔 ( 𝑥 ) = 𝑐 𝑥 + 𝑘
  • D 𝑔 ( 𝑥 ) = 𝑐 𝑥 + 𝑘

Where is the origin mapped to when transforming from the graph of 𝑓 to the graph of 𝑔 ?

  • A ( , 𝑐 𝑘 )
  • B ( , 𝑘 )
  • C ( 𝑘 , )
  • D ( , 𝑘 )

Find 𝑔 ( + 1 ) .

  • A 𝑔 ( + 1 ) = 𝑐 + 𝑘
  • B 𝑔 ( + 1 ) = 𝑐 + 𝑐 𝑘
  • C 𝑔 ( + 1 ) = 𝑐 + 1 𝑐 + 𝑘
  • D 𝑔 ( + 1 ) = 𝑐 + 1 𝑘 +

Use your previous answers to find the equation of the graph shown in the given diagram.

  • A 𝑦 = 2 𝑥 1 + 3
  • B 𝑦 = 2 𝑥 2 + 3
  • C 𝑦 = 2 4 𝑥 4 + 3
  • D 𝑦 = 𝑥 1 + 3

Q18:

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