Worked solution
Reciprocate three supplied sketches
Given the three graphs of , sketch the corresponding graphs of . Indicate, if possible, the coordinates of the turning points, axial intercepts and asymptotes.
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CJC · Transformations · Example 17
(a) Reciprocate the positive bowl
Answer
Local maximum and y-intercept ; horizontal asymptote .
The given bowl has minimum with as drawn, no zeros, and outputs growing without bound at both ends. Its reciprocal stays positive, has no x-intercepts or vertical asymptotes, and tends to zero at both ends. No unique formula for the given curve is inferred.
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(b) Keep the excluded origin
Answer
Vertical asymptote ; horizontal asymptote ; a hole at .
The given graph has horizontal asymptote , vertical asymptote and root , with as drawn. The root produces the reciprocal pole; the finite non-zero horizontal limit is inverted. The original domain excludes 0, so the reciprocal also excludes 0 even though both one-sided limits there are 0. The source labels and draws the curve through it; retain that printed claim in the native/review evidence and show an open hole in the mathematical sketch. There are no actual axial intercepts. No turning point is indicated.
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(c) Transform non-zero turning points and the oblique limit
Answer
Local minimum ; local maximum ; horizontal asymptote and a hole at .
The given sketch has local maximum , local minimum , vertical asymptote and oblique asymptote . Invert the non-zero turning-point heights. At either infinite end, f grows in magnitude along its oblique asymptote, so the reciprocal tends to 0. At the excluded input −1 the reciprocal tends to 0 but remains undefined; the source label is a limit location, not an actual x-intercept. No zero of f is drawn, so no reciprocal vertical asymptote is indicated. The numerical y-intercept cannot be determined from the labelled features alone; no unique formula is inferred.
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