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Lesson Worksheet: Combining Function Transformations Mathematics

In this worksheet, we will practice identifying function transformations that include a combination of translation, dilation, and reflection.

Q1:

The following graph represents a function 𝑔(𝑥) after a reflection on the 𝑥-axis and a horizontal dilation by a scale factor of 12. Which of the following represents the original function 𝑓(𝑥)?

  • A𝑓(𝑥)=4𝑥+2
  • B𝑓(𝑥)=2𝑥+1
  • C𝑓(𝑥)=4𝑥+1
  • D𝑓(𝑥)=2𝑥1
  • E𝑓(𝑥)=4𝑥+2

Q2:

The function 𝑓(𝑥)=(𝑥+1)+2 is reflected in the 𝑥-axis and then shifted 3 units in the positive 𝑦-direction. Write the equation of the transformed function 𝑔(𝑥).

  • A𝑔(𝑥)=(𝑥+1)+1
  • B𝑔(𝑥)=(𝑥+1)+5
  • C𝑔(𝑥)=(𝑥2)+2
  • D𝑔(𝑥)=(𝑥+1)+5
  • E𝑔(𝑥)=(𝑥2)2

Q3:

The following graph represents a function 𝑓(𝑥). Find the resulting function 𝑔(𝑥) after a reflection on the 𝑦-axis followed by a translation by 2 units in the positive direction of the 𝑥-axis.

  • A𝑔(𝑥)=2(𝑥+3)+8
  • B𝑔(𝑥)=(𝑥+1)4
  • C𝑔(𝑥)=(𝑥+5)4
  • D𝑔(𝑥)=(𝑥+1)+4
  • E𝑔(𝑥)=(𝑥+5)4

Q4:

Which of the following processes would you use to obtain the graph of 𝑦=𝑓(3𝑥) from the graph of 𝑦=𝑓(𝑥)?

  1. reflect the graph in the 𝑥-axis
  2. reflect the graph in the 𝑦-axis
  3. stretch the graph in the horizontal direction by a factor of 3
  4. stretch the graph in the horizontal direction by a factor of 13
  • Ac and d
  • Bb and d
  • Ca and c
  • Da and d
  • Eb and c

Q5:

If the following graph is reflected in the 𝑥-axis and then shifted to the left by 1 unit and down by 3 units, what is the equation of the new graph?

  • A𝑓(𝑥)=|𝑥+1|3
  • B𝑓(𝑥)=|𝑥+1|+3
  • C𝑓(𝑥)=|𝑥+1|3
  • D𝑓(𝑥)=|𝑥|3

Q6:

The function 𝑓(𝑥)=(𝑥+2)+3 is stretched in the vertical direction by a scale factor of 2 then reflected in the 𝑦-axis. Write the equation of the transformed function 𝑔(𝑥).

  • A𝑔(𝑥)=(𝑥+4)+3
  • B𝑔(𝑥)=2(𝑥+2)+6
  • C𝑔(𝑥)=(2𝑥+2)+3
  • D𝑔(𝑥)=2(𝑥+2)6
  • E𝑔(𝑥)=(2𝑥+2)3

Q7:

The graph of the function 𝑓(𝑥)=𝑥 is first reflected symmetrically over the 𝑦 axis, then shifted up 2 units and right by 3, and finally horizontally stretched by 2 to obtain the graph of the function 𝑔(𝑥). Write an equation for 𝑔(𝑥).

  • A𝑔(𝑥)=12𝑥3+2
  • B𝑔(𝑥)=12𝑥2+3
  • C𝑔(𝑥)=2𝑥3+2
  • D𝑔(𝑥)=12𝑥+3+2
  • E𝑔(𝑥)=12𝑥3+2

Q8:

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

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

  • A(a)
  • B(b)
  • C(c)

Q9:

The graph represents a function 𝑔(𝑥) after a vertical positive shift by 3 units followed by a reflection on the 𝑥-axis. Which of the following represents the original function 𝑓(𝑥)?

  • A𝑓(𝑥)=3(𝑥1)6
  • B𝑓(𝑥)=(3𝑥1)+2
  • C𝑓(𝑥)=(𝑥1)+2
  • D𝑓(𝑥)=(𝑥4)2
  • E𝑓(𝑥)=(𝑥+2)2

Q10:

The red graph in the figure has the equation 𝑦=𝑓(𝑥) and the blue graph has the equation 𝑦=𝑔(𝑥). Which of the following will not transform the graph of 𝑓(𝑥) onto the graph of 𝑔(𝑥)?

  • AA reflection in the 𝑦-axis, followed by a horizontal shift of 4 units to the left
  • BA vertical shift of 4 units down, followed by a reflection in the 𝑥-axis
  • CA horizontal shift of 4 units to the right, followed by a reflection in the 𝑦-axis
  • DA reflection in the 𝑥-axis

This lesson includes 29 additional questions and 189 additional question variations for subscribers.

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