Sketching Polynomials
1. Sketch from structure, not from plotting many points
A high-quality exam sketch shows defining features: roots, intercepts, asymptotes, turning behaviour and end behaviour.
Cubic end behaviour
\[y=ax^3+\cdots:\ a>0\Rightarrow y\to-\infty\text{ as }x\to-\infty,\ y\to\infty\text{ as }x\to\infty\]Repeated factor
\[(x-r)^2\Rightarrow\text{curve touches the }x\text{-axis at }r\]Reciprocal basic graph
\[y=\frac{k}{x}:\quad x=0,\ y=0\text{ are asymptotes}\]Shifted reciprocal
\[y=\frac{k}{x-h}+q:\quad x=h,\ y=q\text{ are asymptotes}\]- Factorise if possible and mark all real roots.
- Find the y-intercept.
- Determine multiplicity: cross at odd multiplicity; touch at even multiplicity.
- For rational graphs, locate vertical/horizontal asymptotes and forbidden x-values.
- Use leading-term end behaviour and transformations to complete the sketch.
2. Transformations of familiar graphs
Use transformations when the function is built from a known parent graph.
Vertical shift
\[y=f(x)+a\]Horizontal shift
\[y=f(x-a)\]Vertical scale/reflection
\[y=af(x)\]Horizontal scale/reflection
\[y=f(ax)\]Reciprocal transformation
For \(y=\frac{3}{x-2}-4\), the vertical asymptote is \(x=2\) and the horizontal asymptote is \(y=-4\). The sign of 3 determines which pair of opposite “quadrants” relative to the asymptote intersection the branches occupy.Common mistake. Horizontal transformations work inside the function and therefore feel reversed: \(f(x-3)\) moves the graph 3 units right.
3. Using graphs to solve equations
Graph interpretation questions test whether you can translate between equation structure and visual features.
Intersection method
To solve \(f(x)=g(x)\) graphically, draw both curves and read the x-coordinates of their intersections. Equivalently, sketch \(y=f(x)-g(x)\) and read its roots.Exam tip. If the question asks for a sketch only, do not imply false precision. If it asks for an estimate, state the requested accuracy and show which intersection you used.