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)=x2f(x)=x^2
    Describe the transformation:
    1. y=f(x)+2y=f(x)+2
      Solution
    2. y=f(x−3)y=f(x-3)
      Solution
    3. y=−2f(x)+1y=-2f(x)+1
      Solution
    4. y=f(2x)y=f(2x)
      Solution
    5. y=3f(3(x−1))+2y=3f\left(3(x-1)\right)+2
      Solution
  2. f(x)=x2f(x)=x^2
    What is the actual equation of y=2f(3x)−1y=2f(3x)-1?
    Click here if you need to review function substitution.
    Solution
  3. f(x)=x2f(x)=x^2. g(x)=f(2x−6)g(x)=f(2x-6)
    Describe the horizontal translation of g(x)g(x) as compared to f(x)f(x)
    Solution
  4. f(x)=(x−2)(x−6)f(x)=(x-2)(x-6). g(x)=2f(x+1)g(x)=2f(x+1)
    Find the equation of the line of symmetry of g(x)g(x)
    Solution
  5. f(x)=x2f(x)=x^2. g(x)=f(−x)g(x)=f(-x)
    What happens to the point (3,9)(3,9) on f(x)f(x) in this transformation?
    Solution
  6. f(x)=x2f(x)=x^2. g(x)=f(x+3)−1g(x)=f(x+3)-1
    1. Describe this transformation using vector notation
      (pq)\begin{pmatrix}p \\ q \end{pmatrix}
      Solution
    2. Describe this transformation using mapping notation (x,y)→(?,?)(x,y)\to (?,?)
      Solution
  7. Describe the transformation of:
    1. y=f(23x−45)y=f\left(\frac{2}{3}x-\frac{4}{5}\right)
      Solution
    2. h(t)=f(0.2t−π2)+eh(t)=f\left(0.2t-\frac{\pi}{2}\right)+e
      Solution
  8. f(x)=x2f(x)=x^2. g(x)=9f(x)g(x)=9f(x)
    1. h(x)h(x) is a horizontal transformation of f(x)f(x).
      Also h(x)=g(x)h(x)=g(x)
      Then h(x)=f(kx)h(x)=f(kx). Find kk
      Solution
    2. Challenge: Given f(5x)=kf(3x)f(5x)=kf(3x), find kk
      Solution
  9. f(x)=2x+3f(x)=2x+3. g(x)=23f(1−2x)g(x)=\frac{2}{3}f(1-2x)
    What happens to the point (1,5)(1,5) on f(x)f(x) after the transformation?
    Solution
  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)=x2f(x)=x^2
    2. f(x)=2xf(x)=2^x
    3. f(x)=(1/2)xf(x)=(1/2)^x
    4. f(x)=log⁡xf(x)=\log x
    5. f(x)=1xf(x)=\frac{1}{x}
    6. f(x)=sin⁡xf(x)=\sin x
    7. f(x)=cos⁡xf(x)=\cos x
    8. f(x)=tan⁡xf(x)=\tan x
  11. See f(x)f(x) below:

    Sketch y=−2f(x−1)−3y=-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(b(x±c))±dy=af\left(b(x\pm c)\right)\pm d

    Solution
  13. Enrichment: Memorize some additional base functions that can be transformed:
    1. y=⌊x⌋y=\lfloor x \rfloor
      Solution
    2. y=∣x∣xy=\frac{|x|}{x}
      Solution
  14. g(x)=2x−3x+1g(x)=\frac{2x-3}{x+1} and f(x)=1xf(x)=\frac{1}{x}
    1. Describe g(x)g(x) as a transformation of f(x)f(x)
      Solution
    2. Describe the transformation to move the center of g(x)g(x) to the origin
      Solution
  15. Enrichment: Reciprocal transformations
    1. f(x)=2x+3f(x)=2x+3. Sketch g(x)=1f(x)g(x)=\frac{1}{f(x)}
    2. f(x)=(x−2)2−1f(x)=(x-2)^2-1. Sketch g(x)=1f(x)g(x)=\frac{1}{f(x)}
    3. f(x)=x2+4f(x)=x^2+4. Sketch g(x)=1f(x)g(x)=\frac{1}{f(x)}
  16. Enrichment: Given f(x)f(x) below, roughly sketch f(x)\sqrt{f(x)} and state the invariant points
    1. f(x)=x2+4f(x)=x^2+4
    2. f(x)=x2−4f(x)=x^2-4
    3. f(x)=4−x2f(x)=4-x^2
    4. f(x)=−x2−4f(x)=-x^2-4
    5. f(x)=2−xf(x)=2-x
  17. Enrichment: f(x)=x(x−2)f(x)=x(x-2)
    1. Sketch y=∣f(x)∣y=|f(x)|
    2. Sketch y=f(∣x∣)y=f\left(|x|\right)
    3. Sketch y=f(∣x∣−2)y=f\left(|x|-2\right)
    4. Sketch y=−2∣f(x−2)∣y=-2|f(x-2)|
    5. Sketch y=f(∣2x−4∣−2)y=f\left(|2x-4|-2\right)
  18. Challenge: g(x)=2−3x2x−3g(x)=\frac{2-3x}{2x-3}
    Describe g(x)g(x) as a transformation of f(x)=1xf(x)=\frac{1}{x}
    Solution