Rational Functions
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
For each function, state the equations of all asymptotes:
(a)\(f(x)=\dfrac{3}{x+2}\)
(b)\(f(x)=\dfrac{2x-1}{x+4}\)
(c)\(f(x)=\dfrac{x^2+1}{x-1}\)
(a)
Answer
VA: \(x=-2\); HA: \(y=0\)
(b)
Degrees equal: HA at ratio of leading coefficients
Answer
VA: \(x=-4\); HA: \(y=2\)
(c)
Numerator degree > denominator: oblique asymptote
Divide: \(x^2+1=(x-1)(x+1)+2\Rightarrow\) oblique \(y=x+1\)
Answer
VA: \(x=1\); Oblique: \(y=x+1\)
Question 2
Sketch \(y=\dfrac{2}{x-3}+1\), showing all asymptotes and intercepts.
Transformation of \(y= frac{1}{x}\)
VA: \(x=3\); HA: \(y=1\); \(y\)-int: set \(x=0\): \(y=- frac{2}{3}+1= frac{1}{3}\); \(x\)-int: set \(y=0\): \( frac{2}{x-3}=-1\Rightarrow x=1\)
Key points
VA \(x=3\), HA \(y=1\), intercepts \((0, frac{1}{3})\) and \((1,0)\)
Question 3
Find the range of \(f(x)=\dfrac{3x+1}{x-2},\;x
eq2\).
Set \(y= frac{3x+1}{x-2}\) and solve for \(x\)
\(y(x-2)=3x+1\Rightarrow x(y-3)=2y+1\Rightarrow x=\dfrac{2y+1}{y-3}\)
Answer
Range: \(f(x)\in\mathbb{R},\;f(x)
eq3\)
Question 4
Solve \(\dfrac{x+1}{x-2}>\dfrac{x}{x+1}\).
Rearrange to one side
\(\dfrac{x+1}{x-2}-\dfrac{x}{x+1}>0\Rightarrow\dfrac{(x+1)^2-x(x-2)}{(x-2)(x+1)}>0\Rightarrow\dfrac{3x+1}{(x-2)(x+1)}>0\)
Critical values: \(x=-1,- frac{1}{3},2\)
Sign analysis: positive on \(-1
2\) Question 5
The graph of \(y=\dfrac{ax+b}{cx+d}\) passes through \((0,2)\) and \((1,3)\), with a vertical asymptote at \(x=-1\). Find \(a,b,c,d\).
VA at \(x=-1\)
\(c(-1)+d=0\Rightarrow d=c\)
Through \((0,2)\)
\( frac{b}{d}=2\Rightarrow b=2d=2c\)
Through \((1,3)\)
\( frac{a+b}{c+d}= frac{a+2c}{2c}=3\Rightarrow a+2c=6c\Rightarrow a=4c\)
Answer
With \(c=1\): \(a=4,b=2,c=1,d=1\), so \(y=\dfrac{4x+2}{x+1}\)
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