Module 2: Lesson 1

 

Self-Check 2
  1. “Your Turn” on page 68
    1. While this question can be solved in multiple ways, only one method is shown here. Start by identifying the transformations that take place, and then map some key points. Finally, sketch the transformed graph.

      The graph of the function has been transformed from the original graph of in the following ways:
      • a vertical stretch by a factor of 2 about the x-axis
      • a reflection across the x-axis
      • a horizontal translation 3 units to the left
      • a vertical translation 1 unit down

      By taking and mapping the key points of (0, 0), (1, 1), (4, 2), and (9, 3), you should get the following:

       
      (0, 0) → (0, 0) → (0, 0) → (−3, 0) → (−3, −1)
       
      (1, 1) → (1, 2) → (1, −2) → (−2, −2) → (−2, −3)
       
      (4, 2) → (4, 4) → (4, −4) → (1, −4) → (1, −5)
       
      (9, 3) → (9, 6) → (9, −6) → (6, −6) → (6, −7)

      Drawing the sketch should give the following:

       
      This shows the graphs of two functions. The first is y equals the square root of x. The second is the transformed function y equals negative 2 times begin square root x plus 3 end square root minus 1. The graph of the original function passes through the key points (0, 0), (1, 1), (4, 2), and (9, 3). These points map to the key points (-3, -1), (-2, -3), (1, -5), and (6, -7) on the graph of the transformed function.
    2. The domain is {x|x ≥ −3, x ∈ R}.

      The range is {y|y ≤ −1, y ∈ R}.

      The domain is really only affected by the horizontal translation, where the horizontal translation 3 to the left changes the domain from {x|x ≥ 0, x ∈ R} to {x|x ≥ −3, x ∈ R}.

      The range is affected by two of the transformations. The reflection about the y-axis changes the range from {y|y ≥ 0, y ∈ R} to {y|y ≤ 0, y ∈ R}. The translation 1 unit down changes the range from {y|y ≤ 0, y ∈ R} to {y|y ≤ −1, y ∈ R}.



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