Year 12 Functions and relations: Sketching power functions

Power functions are functions of the form $f (x) = a x^{n}$ or closely related variations. They look simple, but they create a wide range of graph shapes depending on whether the exponent is positive, negative, even, odd, or fractional. The key to sketching them well is to focus on symmetry, intercepts, domain restrictions, and behaviour near zero and infinity.

Ask four questions first: is the exponent even or odd, positive or negative, whole or fractional, and what does the coefficient do to the orientation?

Main patterns

Even positive powers, such as $x^{2}$ , are symmetric about the y-axis.
Odd positive powers, such as $x^{3}$ , are symmetric about the origin.
Negative powers introduce asymptotic behaviour and usually exclude zero from the domain.
Fractional powers may restrict the domain depending on whether roots of negative numbers appear.

Worked example 1

Problem: Sketch

f (x) = x^{4}

The exponent is even and positive, so the graph is symmetric about the y-axis.
The function is non-negative for all real x.
The graph passes through $(0, 0)$ .
As $| x |$ becomes large, $x^{4}$ grows quickly upward on both sides.
Compared with $y = x^{2}$ , the graph is flatter near zero and steeper further out.

Answer: the sketch is a y-axis-symmetric curve opening upward with vertex at the origin.

Worked example 2

Problem: Describe the graph of

g (x) = 1 / x^{2}

This is a negative even power, since $g (x) = x^{- 2}$ .
The domain excludes zero because division by zero is impossible.
The graph is symmetric about the y-axis.
The function is always positive for non-zero x.
As $x$ approaches zero, the function grows without bound.
As $| x |$ becomes large, the function approaches zero.

Answer: the graph has two branches above the x-axis, with asymptotes

x = 0

and

y = 0

Worked example 3

Problem: Sketch

h (x) = - x^{3}

and describe its symmetry.

The exponent is odd and positive, so the base shape comes from $y = x^{3}$ .
Odd powers are symmetric about the origin.
The negative coefficient reflects the usual cubic in the x-axis.
The graph still passes through $(0, 0)$ , but now decreases from left to right.
For large negative x, the output becomes large positive. For large positive x, the output becomes large negative.

Answer: the graph is an origin-symmetric cubic that falls from the top-left to the bottom-right.

Reading domain and range from the form

In power-function questions, the formula often tells you domain and range before you draw anything. Positive even powers allow every real x but never produce negative outputs if the coefficient is positive. Negative powers usually remove zero from the domain because division by zero is undefined. Root-style powers can also restrict the domain if a square root of a negative value would be needed. Building this habit means your sketch is not just a picture. It becomes a direct expression of the algebraic restrictions in the rule.

Common traps

Ignoring the domain restriction for negative or fractional powers.
Forgetting the symmetry that comes from even and odd exponents.
Sketching asymptotes incorrectly by making the graph touch them.

Revision focus

Power-function sketches improve quickly if you build a mental library of standard shapes. Once you know the behaviour of $x^{2}$ , $x^{3}$ , $1 / x$ , $1 / x^{2}$ , and simple root functions, many harder sketches become variations on something familiar rather than completely new problems.

It is also useful to compare domain and range at the same time as the sketch. A good graph should make those restrictions visible. If the algebra says zero is excluded or only positive outputs are possible, the sketch should show that clearly.

Practice links

Sketching power functions