Porisma

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Sketch the gradient across a pole and a minimum

The graph of y=f(x)y=f(x) has asymptotes x=2x=2, y=−3y=-3 and minimum (−2,−5)(-2,-5), as shown. Sketch y=f′(x)y=f^{\prime}(x), clearly indicating any asymptotes, turning points and axial intercepts. marks

The supplied graph also labels x-intercepts 1,41,4 and y-intercept −3-3.

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Given function with vertical asymptote x=2 and horizontal asymptote y=−3. The left branch initially falls below −3, reaches minimum (−2,−5), then rises through (0,−3) and (1,0) to positive infinity at 2 from the left. The right branch falls from positive infinity at 2 from the right, crosses (4,0), then tends to −3 from above.
Given function with vertical asymptote x=2 and horizontal asymptote y=−3. The left branch initially falls below −3, reaches minimum (−2,−5), then rises through (0,−3) and (1,0) to positive infinity at 2 from the left. The right branch falls from positive infinity at 2 from the right, crosses (4,0), then tends to −3 from above.
Given function with vertical asymptote x=2 and horizontal asymptote y=−3. The left branch initially falls below −3, reaches minimum (−2,−5), then rises through (0,−3) and (1,0) to positive infinity at 2 from the left. The right branch falls from positive infinity at 2 from the right, crosses (4,0), then tends to −3 from above.

[2009/CJC/Promo/10(c)]

CJC · Review problems · Question 2

Read slopes on each branch before placing the derivative’s zero and asymptotes.

Read the worked solution