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?

  • A𝑓(𝑥)=𝑥4−2𝑥+5
  • B𝑓(𝑥)=𝑥4−2𝑥
  • C𝑓(𝑥)=𝑥4−2𝑥+8
  • D𝑓(𝑥)=𝑥4+2𝑥+8
  • E𝑓(𝑥)=𝑥4+2𝑥+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.

  • A𝑦=−4−|𝑥+1|
  • B𝑦=4|𝑥+1|
  • C𝑦=4−|𝑥+1|
  • D𝑦=4+|𝑥+1|
  • E𝑦=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?

  • A𝑔(𝑥)=√𝑥−3
  • B𝑔(𝑥)=−√𝑥−3
  • C𝑔(𝑥)=−√𝑥+3
  • D𝑔(𝑥)=−√𝑥−3
  • E𝑔(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?

  • A𝑦=(𝑥−3)(2𝑥−7)(6−𝑥)
  • B𝑦=(𝑥−1)(2𝑥−3)(4−𝑥)+2
  • C𝑦=(𝑥+1)(2𝑥+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.

  • A𝑦=13𝑓(3𝑥)
  • B𝑦=13𝑓𝑥3
  • C𝑦=3𝑓𝑥3
  • D𝑦=𝑓𝑥+13+13
  • E𝑦=3𝑓(3𝑥)

Q11:

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

  • A𝑦=𝑓(𝑥)−8
  • B𝑦=−8𝑓(𝑥)
  • C𝑦=𝑓(𝑥+8)
  • D𝑦=𝑓(𝑥−8)
  • E𝑦=𝑓(−8𝑥)

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.

  • A𝑦=12𝑓𝑥2
  • B𝑦=𝑓𝑥2
  • C𝑦=2𝑓𝑥2
  • D𝑦=2𝑓(2𝑥)
  • E𝑦=12𝑓(2𝑥)

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.

  • A𝑦=12𝑓(𝑥)
  • B𝑦=𝑓(𝑥2)
  • C𝑦=𝑓(𝑥+2)
  • D𝑦=2𝑓(𝑥)
  • E𝑦=𝑓(2𝑥)

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).

  • A𝐴=3𝑚4, 𝑏=121−2𝑐𝑚
  • B𝐴=𝑚2, 𝑏=121−2𝑐𝑚
  • C𝐴=𝑚2, 𝑏=121+2𝑐𝑚
  • D𝐴=121+2𝑐𝑚, 𝑏=𝑐𝑚−2
  • E𝐴=2𝑚, 𝑏=1−2𝑐𝑚

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.

  • A𝐴=34, 𝑏=3
  • B𝐴=32, 𝑏=−94
  • C𝐴=23, 𝑏=114
  • D𝐴=32, 𝑏=4
  • E𝐴=43, 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.

  • A𝑦=−13√−𝑥+2−1
  • B𝑦=13√−𝑥+2−1
  • C𝑦=−13√𝑥−2−1
  • D𝑦=−13√−𝑥−2−1
  • E𝑦=−13√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.

  • A𝑦=√−𝑥+1−2
  • B𝑦=−√𝑥+1−2
  • C𝑦=−√𝑥+1+2
  • D𝑦=√−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.

  • A𝑦=√𝑥+1+2
  • B𝑦=−√𝑥+1+2
  • C𝑦=√−𝑥+1+2
  • D𝑦=−√−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?

  • AE
  • BF
  • CH
  • DG

Write its equation.

  • A𝑦=−√𝑥+1−2
  • B𝑦=−√𝑥+1+2
  • C𝑦=√𝑥+1+2
  • D𝑦=√𝑥+1−2

Q20:

Consider the function 𝑓(𝑥)=3√𝑥.

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

  • AD
  • BC
  • CB
  • DA

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

  • Atranslation three units up
  • Bdilation in the horizontal direction by a factor of 3
  • Cdilation in the vertical direction by a factor of 3

State the domain and range of 𝑓(𝑥).

  • Adomain: 𝑥≥0, range: 𝑦≥3
  • Bdomain: 𝑥≥0, range: 𝑦≥0
  • Cdomain: 𝑥≥3, range: 𝑦≥0

Q21:

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).

  • A𝑔(𝑥)=√𝑏𝑥−𝑐
  • B𝑔(𝑥)=𝑏𝑥−𝑐𝑏
  • C𝑔(𝑥)=√𝑐𝑥−𝑏
  • D𝑔(𝑥)=𝑏√𝑥−𝑐

Find 𝑥 so that 𝑔(𝑥)=0.

  • A𝑥=−𝑐𝑏
  • B𝑥=−𝑏𝑐
  • C𝑥=𝑐𝑏
  • D𝑥=𝑏𝑐

Find 𝑥 so that 𝑔(𝑥)=1.

  • A𝑥=1−𝑏𝑐
  • B𝑥=𝑐+1𝑏
  • C𝑥=𝑏+1𝑐
  • D𝑥=1−𝑐𝑏

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

  • A𝑦=2𝑥−12
  • B𝑦=2𝑥−12
  • C𝑦=√2𝑥−1
  • D𝑦=𝑥−12

Q22:

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.

  • A𝑔(𝑥)=√𝑥+5+3
  • B𝑔(𝑥)=√𝑥−5−3
  • C𝑔(𝑥)=√𝑥+3−5
  • D𝑔(𝑥)=√𝑥−5+3
  • E𝑔(𝑥)=√𝑥+5−3

State the domain and range of 𝑔(𝑥).

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

Q23:

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 ℎ units horizontally and 𝑘 units vertically. Write an equation for 𝑔(𝑥).

  • A𝑔(𝑥)=𝑐√𝑥−ℎ+𝑘
  • B𝑔(𝑥)=𝑐√𝑥−ℎ+𝑘
  • C𝑔(𝑥)=𝑐√𝑥−𝑘+ℎ
  • D𝑔(𝑥)=√𝑐𝑥−ℎ+𝑘

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

  • A(𝑘,ℎ)
  • B(ℎ,𝑐𝑘)
  • C(−ℎ,𝑘)
  • D(ℎ,𝑘)

Find 𝑔(ℎ+1).

  • A𝑔(ℎ+1)=𝑐√ℎ+1−𝑘+ℎ
  • B𝑔(ℎ+1)=𝑐+𝑐𝑘
  • C𝑔(ℎ+1)=𝑐+𝑘
  • D𝑔(ℎ+1)=√𝑐ℎ+1−ℎ𝑐+𝑘

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

  • A𝑦=2√𝑥−1+3
  • B𝑦=2√4𝑥−4+3
  • C𝑦=√2𝑥−2+3
  • D𝑦=√𝑥−1+3

Q24:

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

Which of the shown graphs is the graph of 𝑓?

  • AB
  • BF
  • CH
  • DD

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

  • AH
  • BB
  • CD
  • DF

Write its equation.

  • A𝑦=−√𝑥+1+2
  • B𝑦=−√−𝑥+1−2
  • C𝑦=√𝑥+1−2
  • D𝑦=√−𝑥+1−2

Q25:

This is the graph of ℎ(𝑥).

Which of the following is the graph of ℎ𝑥2?

  • A(c)
  • B(a)
  • C(b)

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