Higher Mathematics

Higher Maths Sketching polynomials

Practise sketching polynomials for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Use roots, intercepts, stationary points and end behaviour to sketch polynomial graphs.

Topic practice Formula support Mixed revision

Before you start

Factorise polynomials.
Find y-intercepts.
Recognise repeated roots.

Higher Mathematics lesson

Explanation

A reliable polynomial sketch combines roots, multiplicities, the y-intercept and end behaviour. Calculus can refine the sketch with stationary points.

A graph crosses the x-axis at a simple root and touches it at an even repeated root.

Method and rules

Odd multiplicity: the graph crosses the x-axis.
Even multiplicity: the graph touches the x-axis.
The leading term controls end behaviour for large |x|.

Worked examples

Worked example 1

Identify key sketch features

Describe the key features of y = (x + 2)(x − 1)²

The roots are −2 and 1.
The root 1 is repeated, so the graph touches there.
The leading term is positive x³, so the graph runs from bottom-left to top-right.

Answer: Cross at (-2, 0), touch at (1, 0), with cubic end behaviour.

Worked example 2

Method check

What does a double root do on a polynomial graph?

Its sign does not change at an even root.

So: The graph touches the x-axis and turns.

Watch out

Crossing the axis at a double root.
Ignoring the leading coefficient.
Drawing a sketch without labelled intercepts.

Exam reminder

For sketching polynomials, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.

Continue with the methods that connect most closely to this topic.

Factor and remainder theorem Increasing, decreasing and derivative graphs Stationary points and optimisation All Higher Maths topics Higher Maths worksheets