Year 12: Functions and relations

The Year 12 functions course is about shape, transformation and composition. You are expected to move between an equation, a graph and a description of how one function has been built from another.

Read a graph from the outside in: overall shape first, then intercepts, symmetry, turning points, asymptotes and transformations.

What to master

Sketching polynomial and power functions from key features rather than point-by-point plotting.
Recognising transformations such as translations, dilations and reflections.
Combining functions by sum, product and composition, and interpreting each graphically.
Solving equations created by intersections or composite-function conditions.

Exam habit: when a question asks you to sketch, label the features that justify the sketch. A correct-looking graph without intercepts or turning points marked is incomplete.

How to read a function question

Functions questions often ask for more than one representation at once. You might be given an equation and asked for a graph, or shown a graph and asked to describe the transformation that produced it. The quickest way to stay in control is to move feature by feature. Ask about domain, intercepts, symmetry, turning behaviour, end behaviour, and any translation or reflection. That habit is much better than trying to hold the whole graph in your head at once. It also makes composite-function questions easier, because you start to see how one function acts inside another.

Worked example

Problem: Let

f (x) = x^{2}

and

g (x) = 2 x - 3

. Find

f (g (x))

and describe the effect of

g

before

f

Substitute $g (x)$ into $f$ : $f (g (x)) = {(2 x - 3)}^{2}$ .
If expanded, this is $4 x^{2} - 12 x + 9$ .
Graphically, the inner function $g (x) = 2 x - 3$ changes the input before the square function is applied.
The expression $2 x - 3$ signals a horizontal compression by factor 2 together with a shift linked to the value 3.

Answer:

f (g (x)) = {(2 x - 3)}^{2}

, and the composition changes the input before the squaring happens.

Common traps

Sketching a transformed graph without marking intercepts or turning points.
Confusing a horizontal transformation with a vertical one.
Treating $f (g (x))$ as if it were $f (x) g (x)$ .
Solving composite-function equations without checking the allowable inputs.

Study routine

When revising, try to say each transformation in words before drawing it. For example, "shift right 2, then up 3" or "reflect in the x-axis, then stretch vertically by factor 2". That verbal habit makes your graphing much more deliberate and helps prevent the classic sign mistakes.

Another useful habit is to compare the symbolic and graphical views of the same function. If the algebra says the leading term is positive and of even degree, then the graph should rise on both ends. If a composition changes the input first, the graph should respond horizontally before vertically. Those checks build real fluency between formulas and pictures.

Revision focus

A strong Year 12 habit is to move between representations instead of trusting just one. In algebra, that means checking whether each step preserves the same meaning. In functions and trigonometry, it means linking equations, graphs, intervals, and diagrams. In coordinate geometry, it means asking whether the computed coordinates still match the geometric story in the question. When several representations agree, your answer is usually on solid ground.

It also helps to add one final sentence of interpretation after the working is done. State what the solution means in the language of the problem, not just in symbols. That last step slows you down just enough to catch sign errors, missing restrictions, interval mistakes, or values that are mathematically correct but contextually impossible.