Recover a rational curve from its asymptotes

Original rational fGiven qualitative graph of f, with a vertical asymptote along the y-axis and an oblique asymptote crossing the axes at 3/2 and 3. No asymptote equation is supplied.xy33/2
Question diagram 1
Original rational fGiven qualitative graph of f, with a vertical asymptote along the y-axis and an oblique asymptote crossing the axes at 3/2 and 3. No asymptote equation is supplied.xy33/2

Given graph with oblique-asymptote intercepts3/2 and3; no derived asymptote equation.

The diagram above shows the graph of the function y=f(x)y=f(x) where f(x)=ax2+bx+c+1x+df(x)=\frac{ax^2+bx+c+1}{x+d}, a,b,c,d∈Ra,b,c,d\in\mathbb R. The oblique asymptote intersects the axes at x=32x=\frac32 and y=3y=3. Find the values of a,b,da,b,d.

By considering dydx\frac{dy}{dx}, show that c>−1c>-1. marks

Sketch, on a separate diagram, the graph of y=f′(x)y=f'(x). marks

[2008/NJC/Prelim/I/11(i)]

Total marks: 8

Solution

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  1. Constants Asymptotes of rational functions →
  2. Derivative sketch Gradient graphs →