Transformations Practice Problems

Here you will find a collection of concise function transformation practice problems. Visit this page directly at hunkim.com/transformations

1. f(x)=x^2
   Describe the transformation:
   1. y=f(x)+2
      Solution
      Shift 2 units up
   2. y=f(x-3)
      Solution
      Shift 3 units to the right
   3. y=-2f(x)+1
      Solution
      Multiply y’s by -2, shift 1 unit up
   4. y=f(2x)
      Solution
      Multiply x’s by \frac{1}{2}
   5. y=3f\left(3(x-1)\right)+2
      Solution
      Multiply y’s by 3, multiply x’s by \frac{1}{3}, shift 1 right and 2 up
2. f(x)=x^2
   What is the actual equation of y=2f(3x)-1?
   Click here if you need to review function substitution.
   Solution
   y=18x^2-1
3. f(x)=x^2. g(x)=f(2x-6)
   Describe the horizontal translation of g(x) as compared to f(x)
   Solution
   Shift 3 units right
4. f(x)=(x-2)(x-6). g(x)=2f(x+1)
   Find the equation of the line of symmetry of g(x)
   Solution
   x=-3
5. f(x)=x^2. g(x)=f(-x)
   What happens to the point (3,9) on f(x) in this transformation?
   Solution
   Moves to (-3,9)
6. f(x)=x^2. g(x)=f(x+3)-1
   1. Describe this transformation using vector notation
      \begin{pmatrix}p \\ q \end{pmatrix}
      Solution
      p=-3 and q=-1
   2. Describe this transformation using mapping notation (x,y)\to (?,?)
      Solution
      (x,y)\to(x-3,y-1)
7. Describe the transformation of:
   1. y=f\left(\frac{2}{3}x-\frac{4}{5}\right)
      Solution
      Multiply x’s by \frac{3}{2}, shift \frac{6}{5} to the right
   2. h(t)=f\left(0.2t-\frac{\pi}{2}\right)+e
      Solution
      Multiply x’s by 5, shift \frac{5\pi}{2} right and e units up
8. f(x)=x^2. g(x)=9f(x)
   1. h(x) is a horizontal transformation of f(x).
      Also h(x)=g(x)
      Then h(x)=f(kx). Find k
      Solution
      3
   2. Challenge: Given f(5x)=kf(3x), find k
      Solution
      \frac{25}{9}
9. f(x)=2x+3. g(x)=\frac{2}{3}f(1-2x)
   What happens to the point (1,5) on f(x) after the transformation?
   Solution
   Moves to \left(0,\frac{10}{3}\right)
10. We have mainly focused on transforming quadratics. However all functions can be transformed. Try to memorize the following key base functions:
    1. f(x)=x^2
    2. f(x)=2^x
    3. f(x)=(1/2)^x
    4. f(x)=\log x
    5. f(x)=\frac{1}{x}
    6. f(x)=\sin x
    7. f(x)=\cos x
    8. f(x)=\tan x
11. See f(x) below:
    
    Sketch y=-2f(x-1)-3 on top of the graph above
    Solution
    
    
    
12. Enrichment: Produce the following function using piecewise notation on Desmos and transform the graph using parameter sliders: y=af\left(b(x\pm c)\right)\pm d
    
    Solution
    
    
    
13. Enrichment: Memorize some additional base functions that can be transformed:
    1. y=\lfloor x \rfloor
       Solution
       
       
       
    2. y=\frac{|x|}{x}
       Solution
       
       
       
14. g(x)=\frac{2x-3}{x+1} and f(x)=\frac{1}{x}
    1. Describe g(x) as a transformation of f(x)
       Solution
       Multiply y’s by -5, shift 1 left and 2 up
       
       
    2. Describe the transformation to move the center of g(x) to the origin
       Solution
       Move 1 right, 2 down
       
       
15. Enrichment: Reciprocal transformations
    1. f(x)=2x+3. Sketch g(x)=\frac{1}{f(x)}
    2. f(x)=(x-2)^2-1. Sketch g(x)=\frac{1}{f(x)}
    3. f(x)=x^2+4. Sketch g(x)=\frac{1}{f(x)}
       
16. Enrichment: Given f(x) below, roughly sketch \sqrt{f(x)} and state the invariant points
    1. f(x)=x^2+4
    2. f(x)=x^2-4
    3. f(x)=4-x^2
    4. f(x)=-x^2-4
    5. f(x)=2-x
17. Enrichment: f(x)=x(x-2)
    1. Sketch y=|f(x)|
    2. Sketch y=f\left(|x|\right)
    3. Sketch y=f\left(|x|-2\right)
    4. Sketch y=-2|f(x-2)|
    5. Sketch y=f\left(|2x-4|-2\right)
       
18. Challenge: g(x)=\frac{2-3x}{2x-3}
    Describe g(x) as a transformation of f(x)=\frac{1}{x}
    Solution
    Many correct solutions (verify on Desmos). For example: Multiply y’s by -5, multiply x’s by \frac{1}{4}, shift \frac{3}{2} right and \frac{3}{2} down