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Difficulty: MediumFunction Notation and Transformations

The function gg is defined by g(x)=f(x−3)−2g(x) = f(x - 3) - 2. If the point (5,7)(5, 7) lies on the graph of y=f(x)y = f(x) in the xyxy-plane, which of the following points must lie on the graph of y=g(x)y = g(x)?

  1. A
    (2,9)(2, 9)
  2. (8,5)(8, 5)Answer
  3. C
    (2,5)(2, 5)
  4. D
    (8,9)(8, 9)

Answer

The point (8,5)(8, 5)
The transformation g(x)=f(x−3)−2g(x) = f(x - 3) - 2 shifts the graph of f(x)f(x) to the right by 3 units and down by 2 units. Therefore, the x-coordinate of the point (5,7)(5, 7) increases by 3 (5+3=85 + 3 = 8), and the y-coordinate decreases by 2 (7−2=57 - 2 = 5). This determines that the point (8,5)(8, 5) must lie on the graph of y=g(x)y = g(x).

Step-by-Step Solution

1
Identify the horizontal and vertical transformations applied to the function f(x)f(x) to get g(x)g(x).
The graph of g(x)=f(x−3)−2g(x) = f(x - 3) - 2 is translated horizontally to the right by 3 units and vertically down by 2 units relative to the graph of f(x)f(x).
By function transformation rules, f(x−h)+kf(x - h) + k translates the graph horizontally by hh units (right if h>0h > 0) and vertically by kk units (up if k>0k > 0).
2
Calculate the new x-coordinate for the point on the transformed graph.
The new x-coordinate is 5+3=85 + 3 = 8.
Translating the graph to the right by 3 units increases all x-coordinates of points on the graph by 3.
3
Calculate the new y-coordinate for the point on the transformed graph.
The new y-coordinate is 7−2=57 - 2 = 5.
Translating the graph down by 2 units decreases all y-coordinates of points on the graph by 2.

Key Concept

Function notation and translations in the coordinate plane
Estimated Time:1m 15s
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Topics

Function Notation and Transformations

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