Lesson Worksheet: Function Transformations: Translations Mathematics

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

Q1:

Which of the following is the graph of 𝑓(π‘₯)=√π‘₯βˆ’1?

  • A(c)
  • B(d)
  • C(a)
  • D(b)

Q2:

The function 𝑦=(π‘₯βˆ’1)(2π‘₯βˆ’3)(4βˆ’π‘₯) is translated 2 units in the direction of the positive x-axis. What is the equation of the resulting function?

  • A𝑦=(π‘₯βˆ’3)(2π‘₯βˆ’7)(6βˆ’π‘₯)
  • B𝑦=(π‘₯βˆ’1)(2π‘₯βˆ’3)(4βˆ’π‘₯)+2
  • C𝑦=(π‘₯+1)(2π‘₯+1)(2βˆ’π‘₯)

Q3:

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

  • A𝑦=𝑓(π‘₯)βˆ’8
  • B𝑦=βˆ’8𝑓(π‘₯)
  • C𝑦=𝑓(π‘₯+8)
  • D𝑦=𝑓(π‘₯βˆ’8)
  • E𝑦=𝑓(βˆ’8π‘₯)

Q4:

Consider the function 𝑓(π‘₯)=√π‘₯βˆ’1+2.

Which of the following is the graph of 𝑦=𝑓(π‘₯)?

  • AC
  • BB
  • CA
  • DD

State the domain and range of 𝑓(π‘₯).

  • Adomain: π‘₯β‰₯βˆ’1, range: 𝑦β‰₯2
  • Bdomain: π‘₯β‰₯1, range: 𝑦β‰₯2
  • Cdomain: π‘₯β‰₯1, range: 𝑦β‰₯βˆ’2
  • Ddomain: π‘₯β‰₯βˆ’1, range: 𝑦β‰₯βˆ’2

Q5:

Consider the root function 𝑓(π‘₯)=√π‘₯.

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

  • A𝑔(π‘₯)=√π‘₯+5+3
  • B𝑔(π‘₯)=√π‘₯βˆ’5βˆ’3
  • C𝑔(π‘₯)=√π‘₯+3βˆ’5
  • D𝑔(π‘₯)=√π‘₯βˆ’5+3
  • E𝑔(π‘₯)=√π‘₯+5βˆ’3

State the domain and range of 𝑔(π‘₯).

  • Adomain: π‘₯β‰₯βˆ’5, range: 𝑦β‰₯βˆ’3
  • Bdomain: π‘₯β‰₯βˆ’3, range: 𝑦β‰₯βˆ’5
  • Cdomain: π‘₯β‰₯5, range: 𝑦β‰₯3
  • Ddomain: π‘₯β‰₯βˆ’5, range: 𝑦β‰₯3
  • Edomain: π‘₯β‰₯5, range: 𝑦β‰₯βˆ’3

Q6:

The figure shows the graph of 𝑦=𝑓(π‘₯) and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(π‘₯)βˆ’2.

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

Q7:

The figure shows the graph of 𝑦=𝑓(π‘₯) and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(π‘₯βˆ’1)+4.

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

Q8:

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

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

Q9:

The figure shows the graph of 𝑦=𝑓(π‘₯) and point 𝐴, which is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(π‘₯βˆ’3).

  • A(2,βˆ’2)
  • B(βˆ’1,1)
  • C(2,3)
  • D(5,1)
  • E(2,4)

Q10:

The figure shows the graph of 𝑦=𝑓(π‘₯) and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(π‘₯)+2.

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

Q11:

The function 𝑦=𝑓(π‘₯) is translated four up. Write, in terms of 𝑓(π‘₯), the equation of the translated graph.

  • A𝑦=𝑓(π‘₯)βˆ’4
  • B𝑦=4𝑓(π‘₯)
  • C𝑦=𝑓(4π‘₯)
  • D𝑦=𝑓(π‘₯+4)
  • E𝑦=𝑓(π‘₯)+4

Q12:

Graph (a) shows the curve 𝑦=π‘₯+2π‘₯, which has a point of inflection at the origin. Determine the equation of graph (b), given that it is a translation of graph (a).

  • A𝑦=π‘₯βˆ’3π‘₯+5π‘₯βˆ’7
  • B𝑦=π‘₯βˆ’3π‘₯+5π‘₯βˆ’5
  • C𝑦=π‘₯βˆ’3π‘₯+5π‘₯βˆ’3
  • D𝑦=π‘₯βˆ’3π‘₯+5π‘₯+1
  • E𝑦=π‘₯+3π‘₯+5π‘₯βˆ’5

Q13:

The function 𝑦=𝑓(π‘₯) is translated three to the right. Write, in terms of 𝑓(π‘₯), the equation of the translated graph.

  • A𝑦=𝑓(π‘₯+3)
  • B𝑦=𝑓(π‘₯βˆ’3)
  • C𝑦=𝑓(3π‘₯)
  • D𝑦=𝑓(π‘₯)+3
  • E𝑦=𝑓(π‘₯)βˆ’3

Q14:

Let 𝐻(π‘₯)=π‘₯βˆ’4. For any function 𝑓(π‘₯), what transformation takes the graph of 𝑦=𝑓(π‘₯) to the graph of 𝑦=𝐻(𝑓(π‘₯))?

  • Aa translation right by 4 units
  • Ba translation up by 4 units
  • Ca translation down by 4 units
  • Da translation left by 4 units
  • Ea reflection in the line π‘₯=4

Q15:

The following is the graph of the function 𝑀, which takes a real number π‘₯ and returns its fractional part.

What are the values of 𝑀(4), 𝑀(4.35) and 𝑀(12,345.67)?

  • A0, 0.35, 0.67
  • B0, 4, 12,345
  • C4, 4, 12,345
  • D4, 0.35, 0.67
  • E0, 0.35, 0.67

What are the values of 𝑀(βˆ’0.01) and 𝑀(βˆ’4.35)? Be careful, this is not as straightforward as for positive inputs!

  • Aβˆ’0.01, 4
  • Bβˆ’0.01, 0.65
  • Cβˆ’0.01, βˆ’0.35
  • D0.99, βˆ’0.35
  • E0.99, 0.65

The following is the graph of the function 𝑓.

By considering suitable translations, express 𝑓(π‘₯) in terms of 𝑀(π‘₯).

  • A𝑓(π‘₯)=𝑀(π‘₯βˆ’1)βˆ’1
  • B𝑓(π‘₯)=𝑀(π‘₯+0.5)+0.5
  • C𝑓(π‘₯)=𝑀(π‘₯+1)βˆ’1
  • D𝑓(π‘₯)=𝑀(π‘₯βˆ’0.5)βˆ’0.5
  • E𝑓(π‘₯)=𝑀(π‘₯βˆ’0.5)+0.5

The following is the graph of the triangle function 𝑇.

Use the absolute value function to express 𝑇(π‘₯) in terms of 𝑓(π‘₯).

  • A𝑇(π‘₯)=|𝑓(π‘₯)|
  • B𝑇(π‘₯)=βˆ’|𝑓(π‘₯)|
  • C𝑇(π‘₯)=|𝑓(π‘₯+0.5)|
  • D𝑇(π‘₯)=βˆ’|𝑓(π‘₯+0.5)|
  • E𝑇(π‘₯)=|𝑓(1βˆ’π‘₯)|

Q16:

The red graph in the figure has equation 𝑦=𝑓(π‘₯) and the blue graph has equation 𝑦=𝑔(π‘₯). Express 𝑔(π‘₯) as a transformation of 𝑓(π‘₯).

  • A𝑔(π‘₯)=𝑓(π‘₯+2)βˆ’2
  • B𝑔(π‘₯)=𝑓(π‘₯+2)
  • C𝑔(π‘₯)=𝑓(π‘₯)+2
  • D𝑔(π‘₯)=𝑓(π‘₯βˆ’2)
  • E𝑔(π‘₯)=𝑓(π‘₯)βˆ’2

Q17:

The figure shows the graph of 𝑦=𝑓(π‘₯).

Which of the following is the graph of 𝑦=𝑓(π‘₯βˆ’2)+3?

  • A
  • B
  • C
  • D
  • E

Q18:

Which of the following equations could be that of the given graph?

  • A𝑦=√π‘₯+1
  • B𝑦=√π‘₯βˆ’1
  • C𝑦=√π‘₯βˆ’1
  • D𝑦=√π‘₯βˆ’1
  • E𝑦=√π‘₯+1

Q19:

Which of the graphs matches the equation 𝑦=(π‘₯βˆ’3)+5?

  • AC
  • BD
  • CA
  • DB

Q20:

The curve of the function 𝑔(π‘₯)=βˆ’5βˆ’|π‘₯| is the same curve of the function 𝑓(π‘₯)=βˆ’|π‘₯| by translation of magnitude 5Β units in which direction?

  • A𝑂π‘₯β€²
  • B𝑂𝑦′
  • Cοƒͺ𝑂𝑦
  • Dοƒͺ𝑂π‘₯

Q21:

The red graph in the figure has the equation 𝑦=𝑓(π‘₯) and the blue graph has the equation 𝑦=𝑔(π‘₯). Express 𝑔(π‘₯) as a transformation of 𝑓(π‘₯).

  • A𝑔(π‘₯)=𝑓(π‘₯)βˆ’4
  • B𝑔(π‘₯)=𝑓(4π‘₯)
  • C𝑔(π‘₯)=4𝑓(π‘₯)
  • D𝑔(π‘₯)=𝑓(π‘₯+4)
  • E𝑔(π‘₯)=𝑓(π‘₯)+4

Q22:

The red graph in the figure has the equation 𝑦=𝑓(π‘₯) and the blue graph has the equation 𝑦=𝑔(π‘₯). Express 𝑔(π‘₯) as a transformation of 𝑓(π‘₯).

  • A𝑔(π‘₯)=𝑓(π‘₯βˆ’2)
  • B𝑔(π‘₯)=𝑓(βˆ’2π‘₯)
  • C𝑔(π‘₯)=βˆ’2𝑓(π‘₯)
  • D𝑔(π‘₯)=𝑓(π‘₯)+2
  • E𝑔(π‘₯)=𝑓(π‘₯)βˆ’2

Q23:

The figure shows the graph of 𝑦=𝑓(π‘₯).

Which of the following is the graph of 𝑦=𝑓(π‘₯+1)?

  • A
  • B
  • C
  • D
  • E

Q24:

The figure shows the graph of 𝑦=𝑓(π‘₯).

Which of the following is the graph of 𝑦=𝑓(π‘₯)+2?

  • A
  • B
  • C
  • D
  • E

Q25:

The function 𝑦=𝑓(π‘₯) is translated five to the left. Write, in terms of 𝑓(π‘₯), the equation of the translated graph.

  • A𝑦=𝑓(π‘₯βˆ’5)
  • B𝑦=𝑓(π‘₯)+5
  • C𝑦=5𝑓(π‘₯)
  • D𝑦=𝑓(π‘₯+5)
  • E𝑦=𝑓(π‘₯)βˆ’5

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