Year 12 Functions and relations: Sketching polynomial functions

Sketching polynomial functions is not about plotting a long table of values. In Year 12, the goal is to read the structure of the polynomial and turn that structure into a graph with the correct shape. The main features come from the degree, the leading coefficient, the intercepts, repeated roots, and any turning behaviour you can infer or calculate. A well-drawn sketch shows that you understand the function qualitatively, not just numerically.

Start with the biggest structural features first: end behaviour, intercepts, multiplicity of roots, and only then the finer graph details.

Main features to identify

The degree tells you the general family of shapes you should expect.
The leading coefficient tells you whether the graph rises or falls on the ends.
The x-intercepts come from factoring or solving the polynomial equation.
A repeated root usually means the graph touches the axis and turns instead of crossing.

Worked example 1

Problem: Sketch the graph of

f (x) = (x - 2) (x + 1) (x - 3)

This is a cubic polynomial, so the graph has odd-degree end behaviour.
The leading term is $x^{3}$ , which has a positive coefficient.
So the graph falls to the left and rises to the right.
The x-intercepts are at $x = - 1$ , $x = 2$ , and $x = 3$ .
Each factor appears once, so each root has multiplicity 1 and the graph crosses the axis at each intercept.
The y-intercept is $f (0) = (- 2) (1) (- 3) = 6$ , so the graph passes through $(0, 6)$ .

Answer: the sketch is a positive cubic crossing the x-axis at

- 1

2

, and

3

, with y-intercept

6

Worked example 2

Problem: Describe the graph of

g (x) = {(x - 1)}^{2} (x + 2)

The degree is 3, so this is still a cubic.
The leading coefficient is positive, so the graph falls left and rises right.
The roots are $x = 1$ and $x = - 2$ .
The root $x = 1$ has multiplicity 2, so the graph touches the x-axis there and turns.
The root $x = - 2$ has multiplicity 1, so the graph crosses the x-axis there.
The y-intercept is $g (0) = {(- 1)}^{2} (2) = 2$ .

Answer: the graph is a positive cubic that crosses at

x = - 2

, touches at

x = 1

, and passes through

(0, 2)

Common traps

Marking a repeated root as a crossing point instead of a turning point.
Ignoring the leading coefficient and getting the end behaviour backwards.
Sketching a correct-looking curve but forgetting to label key intercepts.

Revision focus

A strong polynomial sketch begins with a feature list. Write down the degree, the end behaviour, the roots, the root multiplicities, and the y-intercept before you draw anything. Once those facts are on the page, the graph itself is much easier to place with confidence.

This topic is also a good reminder that algebra and graphing are not separate skills. Factorising the polynomial gives you graph information directly. Reading the graph carefully then tells you whether your algebraic work is plausible. That two-way check is one of the most useful habits in the whole Functions course.

Practice links

Sketching polynomial functions