Worksheet: Transformations of Functions

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

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 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

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(a)
  • C(b)

Q4:

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)

Q5:

Which of the following is the graph of 𝑓 ( 𝑥 ) = 𝑥 1 3 ?

  • A (c)
  • B (b)
  • C (d)
  • D (a)

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

Q8:

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

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

  • A(c)
  • B(a)
  • C(b)

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

Q10:

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

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

Q13:

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 )

Q14:

Fares thinks that he can map the graph of the function to the graph of any other linear function by a translation followed by a stretch. This is equivalent to saying that any linear function can be written in the form for suitable values of and .

Is he right?

  • Ano
  • Byes

Suppose that and is as before. Find and in terms of and when it is possible to write in the form .

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Let and . Find the values of and for which . Use the following graph to help you.

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

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 = 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

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
  • 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

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
  • 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

Q18:

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

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