Module 6: Lesson 1

 

Self-Check 3

  1. “Your Turn” from “Example 1” on page 351 of the textbook

    The transformed graph looks as follows:

     
    This graph shows a curve of y equals 4 exponent x. The graph curves upward, with a horizontal asymptote through x = 0 and a y-intercept at (0, 1). The image also shows the transformed graph after all of the transformations have taken place.

    The graph has undergone the following transformations:
    • Because a = 1, there has been no vertical stretch nor has there been a reflection in the x-axis.
    • Because b = −2, there has been a horizontal stretch by a factor of about the y-axis and a reflection in the y-axis.
    • Because h = −5, there has been a horizontal translation of 5 units to the left.
    • Because k = −3, there has been a vertical translation of 3 units down.

      y = 4x

      y = 4−2x

      y = 4−2(x+5)

      y = 4−2(x+5)−3

      (0, 1)

      (0, 1)

      (−5, 1)

      (−5, −2)

      (1, 4)

      (2, 16)

      (−1, 16)

      (−6, 16)

      (−6, 13)


    The domain in the original graph was {x|x ∈ R}. There has been no change to the domain.

    The range in the original graph was {y|y > 0, y ∈ R}. The vertical translation has the effect of shifting the entire range down 3 units. Thus, the new domain is: {y|y > −3, y ∈ R}.

    The horizontal asymptote in the original graph was y = 0. Like for the range, the vertical translation has the effect of shifting the horizontal asymptote down 3 units. Thus, the new horizontal asymptote is y = −3.

    The original graph had no x-intercept. Because of the vertical translation, there is now an x-intercept (one that can be calculated by letting y = 0 in the transformed equation). The new x-intercept occurs at approximately (−5.4, 0).

    The original graph had a y-intercept of (0, 1). Because of the horizontal translation, the reflection in the y-axis, and the vertical translation, there is a new y-intercept (one that can be calculated by letting x = 0 in the transformed equation). The new x-intercept occurs at approximately (0, −2.999 99). Note that it is not (0, −3) because that is the horizontal asymptote, but it is close to that value.


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