Answer:
C) g(x) = 7|-x -5| +11
Explanation:
You want to transform f(x) = |x -5| +2 with a vertical stretch by a factor of 7, reflection over the y-axis, and translation 3 units down.
Stretch
A vertical stretch is accomplished by multiplying the function value by the stretch factor. A vertical stretch by a factor of 7 looks like ...
g(x) = 7·f(x)
Reflection
Reflection over the y-axis changes the sign of the x-coordinates. The reflected function looks like ...
g(x) = f(-x)
Translation
A vertical translation adds the translation amount to the function value. Translation down by 3 units looks like ...
g(x) = f(x) -3
Composition
The combined effect of these transformations is ...
g(x) = 7·f(-x) -3
g(x) = 7(|-x -5| +2) -3 . . . . . . use the given expression for f(x)
g(x) = 7|-x -5| +11 . . . . . matches choice C
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Additional comment
The function f(x) has its vertex at (5, 2). When we stretch the function vertically, the y-coordinate of the vertex moves to 7·2 = 14. That is, the original translation upward by 2 units is also stretched. Choice B neglects to stretch that original vertical offset.
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