Exercises


  1. The following functions have undergone the following transformations:

    • a vertical stretch about the \(x\)-axis by a factor of \( 2 \);
    • a horizontal stretch about the \(y\)-axis by a factor of \( 3 \);
    • a reflection about the \( x \)-axis; and
    • a translation right \( 9 \) units and down \( 5 \) units.

    Complete the table below:

    Base Function Equation Transformed Function Equation
    (in simplest form)
    \( y = \sqrt{x} \)  
    \( y = \dfrac{1}{x} \)  
  2. Piecewise linear function through (-6,3),(-4,-2),(0,4),(2,2),(4,4),(6,-3), in order from left to rightGiven the graph of the function \( y = f(x) \), draw the graphs of the following transformed functions:
    1. \( y = -f(x) + 3 \)
    2. \( y = 2f\left( -(x - 4) \right) - 2 \)
    3. \( y = \frac{1}{2}f\left( 2(x + 3) \right) \)
  3. State the domain and range of the following functions:
    1. \( h(x) = 2(x + 1)^2 - 3 \)
    2. \( h(x) = -3\sqrt{x + 2} + 5 \)
    3. \( h(x) = \dfrac{4}{3 - x} - 6 \)
    4. \( h(x) = -2^{x - 4} + 1 \)
  4. Write the transformations that were applied to take the graph of \( y = \left \lvert x \right \rvert \) to \( y = -\frac{1}{7}\left \lvert -2x + 6 \right \rvert - 8 \).
  5. The graph of each of the following functions is a transformation of one of the five base functions. Identify a possible equation for the transformed functions.
    1. Graph of y=abs(x) translated right 4 units, up 3 units
    2. Graph of y=sqrt(x) reflected about y-axis, translated right 3 units, down 4 units
    3. Graph of y=x^2 reflected about x-axis, stretched by factor of 3, translated left 2 units, down 4 units
    4. Graph of y=1/x translated right 3 units, up 3 units
  6. Determine a possible set of transformations that can be applied to the graph of \( y = x^2 \) to obtain the graph of \( y = 5 - 2x - 4x^2 \).