Transformations

Basic Functions and Transformations

  1. \displaystyle f(x)=2x+3
    1. Sketch this graph.
    2. Evaluate \displaystyle f(2)
  2. \displaystyle f(x)=\sqrt{x-2}
    1. Sketch this graph and label 2 points.
    2. Find \displaystyle {{f}^{-1}}(x)
    3. Domain and range of \displaystyle {{f}^{-1}}(x)?
  3. \displaystyle g(x)=f(x+2)-1.  \displaystyle f(x)={{x}^{2}}
    1. Describe the transformation.
    2. What is the actual equation of \displaystyle g(x)?
  4. \displaystyle g(x)=f(-x+2).  Describe this transformation.
  5. Reflect the line \displaystyle f(x)=\frac{x}{2}-1 about the x-axis.
    1. What is the new equation?
    2. Express the new function \displaystyle g(x) as a transformation of \displaystyle f(x) using function notation (as opposed to stating the actual equation of \displaystyle g(x)).
    3. Describe this transformation using mapping notation:  \displaystyle (x,y)\to ?
  6. \displaystyle g(x)=\sqrt{x}
    1. Sketch \displaystyle f(x)=g\left( -\frac{4}{9}x \right)
    2. What is the actual equation of \displaystyle f(x)?
    3. Express \displaystyle f(x) as a vertical transformation of \displaystyle g(x)
  7. \displaystyle f(x)={{x}^{2}}.  \displaystyle g(x)=f(5x-3)+1
    1. Describe the transformation.
    2. Coordinate of vertex?
    3. Domain and range?
    4. Sketch this graph and label two points.
  8. \displaystyle h(t)={{t}^{2}}-4t
    1. Find \displaystyle {{h}^{-1}}(t)
    2. Evaluate \displaystyle {{h}^{-1}}(2)
  9. If \displaystyle (1,-2) is on \displaystyle f(x), what point must be on \displaystyle 2{{f}^{-1}}(-x+1)?
  10. Find \displaystyle {{f}^{-1}}(2) if \displaystyle f(3)=2
  11. Enter \displaystyle y={{(x+1)}^{2}}-1\{x\ge -1\} into www.desmos.com
    1. What is the domain and range of this function?
    2. Now graph the inverse of this function.
    3. How are the domain and ranges of \displaystyle f(x) and \displaystyle {{f}^{-1}}(x) related?
    4. What are the invariant points? (going from \displaystyle f(x) to \displaystyle {{f}^{-1}}(x))
    5. \displaystyle {{f}^{-1}}(x) always reflects \displaystyle f(x) along what line?
  12. Sketch \displaystyle f(x)=\left| x(x-4) \right|+2
  13. The graph below is \displaystyle f(x).

    1. Sketch \displaystyle g(x)={{f}^{-1}}(x) on the same graph.
    2. What is the domain and range of \displaystyle f(x)?
    3. What is the domain and range of \displaystyle {{f}^{-1}}(x)?
    4. Evaluate \displaystyle f(0)
    5. For what values of x does \displaystyle f(x)={{f}^{-1}}(x)?
  14. \displaystyle f(x)=\left| x \right|.  See \displaystyle g(x) below:

    1. Express \displaystyle g(x) as a transformation of \displaystyle f(x) using transformation notation:  \displaystyle g(x)=af(b(x\pm c))\pm d
    2. What is the actual equation of \displaystyle g(x)?
  15. Given \displaystyle f(x) below, sketch \displaystyle {{f}^{-1}}(x).

  16. See \displaystyle f(x) below.  Sketch \displaystyle y=2\left| f(-x) \right|

  17. \displaystyle f(x)={{x}^{2}}-2x.  \displaystyle g(x)=\sqrt{f(x+2)}
    1. Sketch \displaystyle g(x)
    2. Coordinates of invariant points? (comparing \displaystyle f(x+2) vs. \displaystyle g(x))
  18. \displaystyle f(x)=\sqrt{x}.  \displaystyle g(x)=f(x+4).  \displaystyle h(x)=x-2.  Where does \displaystyle g(x) and \displaystyle h(x) intersect?  \displaystyle (x,y)
  19. \displaystyle f(x)={{x}^{2}}+4.  \displaystyle g(x)=\sqrt{f(x)}.  Sketch \displaystyle g(x) and label two points.
  20. \displaystyle h(t)=-2(t+4).  \displaystyle z(t)=\sqrt{h(t)}.  Sketch \displaystyle z(t) and label the invariant points.