Sketching Graphs
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
For each function, find: (i) \(x\)- and \(y\)-intercepts, (ii) any asymptotes, (iii) stationary points. Then sketch.
(a)\(f(x)=x^3-3x^2\)
(b)\(f(x)=\dfrac{x}{x^2-1}\)
(a)
Intercepts
\(x\)-int: \(x=0,3\); \(y\)-int: \((0,0)\)
Stationary points
\(f'(x)=3x^2-6x=3x(x-2)=0\Rightarrow x=0\) (local max, \(f=0\)), \(x=2\) (local min, \(f=-4\))
Key features
Local max \((0,0)\); local min \((2,-4)\); cubic shape
(b)
Intercepts
\(x\)-int and \(y\)-int both at origin
Asymptotes
VA: \(x=\pm1\); HA: \(y=0\)
Stationary points
\(f'(x)=\dfrac{(x^2-1)-x(2x)}{(x^2-1)^2}=\dfrac{-(x^2+1)}{(x^2-1)^2}<0\) always — no stationary points
Key features
Three branches; odd function; decreasing throughout each branch
Question 2
Sketch \(y=xe^{-x}\), labelling any intercepts, asymptotes, and the local maximum.
Intercept
\(x=0\): \(y=0\)
Asymptote
As \(x o\infty\): \(y o0\); HA \(y=0\)
Stationary point
\(f'(x)=e^{-x}-xe^{-x}=e^{-x}(1-x)=0\Rightarrow x=1\); \(y=e^{-1}\)
Key features
Origin, local max at \((1,e^{-1})\), HA \(y=0\), negative for \(x<0\)
Question 3
Sketch \(y=\ln(x-2)\), stating domain, range, asymptote, and intercept.
Translation of \(y=\ln x\), 2 right
Answer
Domain: \(x>2\); Range: \(\mathbb{R}\); VA: \(x=2\); \(x\)-int: \((3,0)\)
Question 4
Sketch \(y=\dfrac{x^2-4}{x^2-1}\), showing all asymptotes and intercepts.
Simplify / features
HA: \(y=1\) (leading coefficients); VA: \(x=\pm1\); \(x\)-int: \((\pm2,0)\); \(y\)-int: \((0,4)\)
Behaviour at asymptotes
Check signs in each region
Sketch features
HA \(y=1\), VA \(x=\pm1\), \(x\)-ints \(\pm2\), \(y\)-int 4
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