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Difficulty: MediumCoordinate Geometry: Transformations and Geometric Graphs

In the xyxy-plane, the graph of a function g(x)g(x) is obtained by shifting the graph of f(x)=x2−6x+2f(x) = x^2 - 6x + 2 horizontally to the right by 33 units and then reflecting the resulting graph across the xx-axis. What is the value of g(1)g(1)?

  1. −18-18Answer
  2. B
    −10-10
  3. C
    00
  4. D
    66
  5. E
    1818

Answer

The value of g(1)g(1) is −18-18.
Shifting the graph of f(x)f(x) horizontally to the right by 33 units gives the function h(x)=f(x−3)h(x) = f(x - 3). Reflecting this graph across the xx-axis negates the entire function output, yielding g(x)=−h(x)=−f(x−3)g(x) = -h(x) = -f(x - 3). To find g(1)g(1), evaluate −f(1−3)=−f(−2)-f(1 - 3) = -f(-2). Substituting x=−2x = -2 into f(x)=x2−6x+2f(x) = x^2 - 6x + 2 gives f(−2)=(−2)2−6(−2)+2=4+12+2=18f(-2) = (-2)^2 - 6(-2) + 2 = 4 + 12 + 2 = 18. Negating this gives −18-18.

Step-by-Step Solution

1
Determine the algebraic expression for the transformed function g(x)g(x) in terms of f(x)f(x).
g(x)=−f(x−3)g(x) = -f(x - 3)
Shifting a graph horizontally to the right by 33 units replaces xx with x−3x - 3, giving f(x−3)f(x - 3). Reflecting across the xx-axis negates the entire function, resulting in −f(x−3)-f(x - 3).
2
Substitute x=1x = 1 into the expression for g(x)g(x).
g(1)=−f(1−3)=−f(−2)g(1) = -f(1 - 3) = -f(-2)
Evaluating g(1)g(1) requires finding the value of −f(x−3)-f(x - 3) when x=1x = 1.
3
Evaluate f(−2)f(-2) using the original definition f(x)=x2−6x+2f(x) = x^2 - 6x + 2.
f(−2)=(−2)2−6(−2)+2=4+12+2=18f(-2) = (-2)^2 - 6(-2) + 2 = 4 + 12 + 2 = 18
Substituting −2-2 into f(x)f(x) follows standard algebraic order of operations.
4
Apply the outer negative sign to calculate g(1)g(1).
g(1)=−18g(1) = -18
Since g(1)=−f(−2)g(1) = -f(-2) and f(−2)=18f(-2) = 18, g(1)=−18g(1) = -18.

Key Concept

Graph Transformations of Functions
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Topics

Coordinate Geometry: Transformations and Geometric Graphs

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