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Explain how the graph of f(x)=(-2)/(x-3)^2 can be obtained from the graph of y=1/x^2 by means of translations, compressions, expansions, or reflections

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Tamara J. | Math Tutoring - Algebra and Calculus (all levels)Math Tutoring - Algebra and Calculus (al...
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(1) Translations:     f(x+c) => translates graph c units to the left

f(x-c) => translates graph c units to the right

f(x) = 1/x2     ==>     f(x-3) = 1/(x-3)2     ==>     graph translated 3 units to the right

(2) Reflections:     -f(x)  =>  reflects graph about the x-axis

f(-x)  =>  reflects graph about the y-axis

f(x-3) = 1/(x-3)2     ==>     -f(x-3) = -1/(x-3)2     ==>     graph reflected about the x-axis

(3) Expansions/Compressions:     c·f(x)  =>  if c>1, graph stretches in the y-direction
if 0<c<1, graph is compressed (in y-direction)

f(cx)  =>  if c>1, graph compresses in the x-direction
if 0<c<1, graph is stretched (in x-direction)

-f(x-3) = -1/(x-3)2   ==>   2·(-f(x-3)) = -2f(x-3) = -2/(x-3)2   ==>   graph stretches in y-directions

So, from     f(x)= 1/x2    to     -2f(x-3) = -2/(x-3)2
the graph of the original function is shifted (translated) 3 units to the right, flipped (reflected) about the x-axis, and stretched (expanded) in the y-direction.