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Answer:
1a. translation right 5, down 4
1b. vertical stretch ×3, translation left 3, up 2
2a. translation left 2, down 2
2b. reflect in x, vertical stretch ×2, translation up 3
3. f(x) = -4/(x+5)+4
4. f(x) = 1/2√(3(x +3)) -4
Step-by-step explanation:
Transformations of functions can be decoded by considering the transformation ...
g(x) = a·f((x-h)/b) +k
The transformations here are ...
- vertical stretch by a factor of 'a'
- horizontal stretch by a factor of 'b'
- right shift by h units
- up shift by k units
Note that either 'a' or 'b' can be a fraction less than 1, which results in a transformation usually described as a compression (not a stretch). If either of these is negative, then a reflection over the x-axis (a<0) or y-axis (b<0) is also involved. The stretch (or compression) and/or reflection is applied before the translation the way this is written.
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1a. translation right 5, down 4
1b. vertical stretch by a factor of 3, translation left 3, up 2
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2a. translation left 2, down 2
2b. reflected across the x-axis, vertically stretched by a factor of 2, translated up 3
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3. a=4, b=-1, h=-5, k=4
f(x) = -4/(x+5)+4
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4. a=1/2, b=1/3, h=-3, k=-4
f(x) = 1/2√(3(x +3)) -4
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The graphs in the attachments show the various functions. (It takes a little work to verify that the graph in the last attachment has the appropriate compressions, but it does.)
