Lesson Worksheet: Function Transformations: Dilation Mathematics

In this worksheet, we will practice identifying function transformations involving horizontal and vertical stretches or compressions.

Q1:

The figure shows the graph of 𝑦=𝑓(𝑥).

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

  • A
  • B
  • C
  • D
  • E

Q2:

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.

  • A1,12
  • B(2,1)
  • C(2,2)
  • D2,12
  • E(4,2)

Q3:

The figure shows the graph of 𝑦=𝑓(𝑥).

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

  • A
  • B
  • C
  • D
  • E

Q4:

The figure shows the graph of 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=𝑓𝑥2?

  • A
  • B
  • C
  • D
  • E

Q5:

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

  • A𝑔(𝑥)=12𝑓(2𝑥)
  • B𝑔(𝑥)=𝑓(2𝑥)
  • C𝑔(𝑥)=2𝑓(𝑥)
  • D𝑔(𝑥)=𝑓𝑥2
  • E𝑔(𝑥)=12𝑓(𝑥)

Q6:

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=2𝑓(𝑥).

  • A(2,2)
  • B(4,1)
  • C2,12
  • D(2,1)
  • E(4,2)

Q7:

The function 𝑦=𝑓(𝑥) is stretched in the vertical direction by a scale factor of 12. Write, in terms of 𝑓(𝑥), the equation of the transformed function.

  • A𝑦=𝑓(𝑥)2
  • B𝑦=2𝑓(𝑥)
  • C𝑦=𝑓(2𝑥)
  • D𝑦=𝑓(𝑥)+2
  • E𝑦=𝑓12𝑥

Q8:

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

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

Q9:

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(2𝑥).

  • A(1,1)
  • B(4,2)
  • C(2,1)
  • D(4,1)
  • E1,12

Q10:

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

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

Q11:

The figure shows the graph of 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=12𝑓(𝑥)?

  • A
  • B
  • C
  • D
  • E

Q12:

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

  • A𝑔(𝑥)=𝑓(𝑥)2
  • B𝑔(𝑥)=𝑓𝑥2
  • C𝑔(𝑥)=2𝑓(𝑥)
  • D𝑔(𝑥)=12𝑓(2𝑥)

Q13:

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

  • A𝑔(𝑥)=𝑓(2𝑥)
  • B𝑔(𝑥)=12𝑓(𝑥)
  • C𝑔(𝑥)=2𝑓(𝑥)
  • D𝑔(𝑥)=𝑓𝑥2
  • E𝑔(𝑥)=12𝑓(2𝑥)

Q14:

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=𝑓𝑥2.

  • A(1,1)
  • B1,12
  • C(4,2)
  • D(4,1)
  • E(2,1)

Q15:

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

  • A𝑔(𝑥)=12𝑓(2𝑥)
  • B𝑔(𝑥)=12𝑓(𝑥)
  • C𝑔(𝑥)=𝑓𝑥2
  • D𝑔(𝑥)=𝑓(2𝑥)
  • E𝑔(𝑥)=2𝑓(𝑥)

Q16:

A car factory manufactures a different number of cars every month based on a certain function 𝑓(𝑥). In a certain year, the factory manager decided to increase the monthly rate by a factor of 2 based on a function 𝑔(𝑥). Find 𝑔(𝑥) in terms of 𝑓(𝑥).

  • A𝑔(𝑥)=𝑓𝑥2
  • B𝑔(𝑥)=2𝑓(𝑥)
  • C𝑔(𝑥)=2𝑓𝑥2
  • D𝑔(𝑥)=𝑓(2𝑥)
  • E𝑔(𝑥)=12𝑓(𝑥)

Q17:

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)

Q18:

The figure shows the graph of 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=2𝑓𝑥2?

  • A
  • B
  • C
  • D
  • E

Q19:

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).

  • A𝑔(𝑥)=2𝑓𝑥2
  • B𝑔(𝑥)=12𝑓(2𝑥)
  • C𝑔(𝑥)=2𝑓(𝑥)
  • D𝑔(𝑥)=𝑓𝑥2
  • E𝑔(𝑥)=12𝑓𝑥2

Q20:

The graphs of three square root functions are shown in the given diagram. The equations for curves A and B have the form 𝑦=√𝑥+𝑝+𝑞 for some integers 𝑝 and 𝑞.

Curve C can be obtained by dilating curve B in the vertical direction. Write the corresponding equation for curve C in the form 𝑦=𝑎√𝑥−ℎ+𝑘.

  • A𝑦=12√𝑥−4−1
  • B𝑦=1√2√𝑥−1−4
  • C𝑦=1√2√𝑥−4−1
  • D𝑦=1√2√𝑥+4−1
  • E𝑦=12√𝑥+4−1

Curve C can also be obtained by dilating curve A in the horizontal direction. Write the corresponding equation for curve C in the form 𝑦=√𝑏𝑥−𝑐+𝑘.

  • A𝑦=12𝑥−1−1
  • B𝑦=1√2𝑥−2−1
  • C𝑦=12𝑥−2−1
  • D𝑦=√𝑥−4−1
  • E𝑦=√𝑥−2−1

Q21:

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

Q22:

The figure shows the graph of 𝑦=𝑓(𝑥).

Which of the following is the graph of 𝑦=12𝑓(2𝑥)?

  • A
  • B
  • C
  • D
  • E

Q23:

This is the graph of ℎ(𝑥).

Which of the following is the graph of ℎ𝑥2?

  • A(c)
  • B(a)
  • C(b)

Q24:

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)

Q25:

The graph of 𝑓(𝑥) is given by the dashed red line in the following figure. Also shown are graphs of 𝑓𝑥2, 𝑓(2𝑥), and 2𝑓(𝑥). Match up these functions to their graphs.

  • A𝑓𝑥2⟷ (a), 2𝑓(𝑥)⟷ (b), 𝑓(2𝑥)⟷ (c)
  • B𝑓𝑥2⟷ (a), 2𝑓(𝑥)⟷ (c), 𝑓(2𝑥)⟷ (b)
  • C𝑓𝑥2⟷ (c), 2𝑓(𝑥)⟷ (b), 𝑓(2𝑥)⟷ (a)
  • D𝑓𝑥2⟷ (c), 2𝑓(𝑥)⟷ (a), 𝑓(2𝑥)⟷ (b)
  • E𝑓𝑥2⟷ (b), 2𝑓(𝑥)⟷(c), 𝑓(2𝑥)⟷ (a)

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