Power functions y = a · xⁿ

What is it about?

A power function has the form

f(x) = a \cdot x^n \qquad (a \neq 0,\; n \text{ an integer})

The exponent $n$ decides the shape of the graph. The stretch factor $a$ stretches ( $|a| > 1$ ) or compresses ( $|a| < 1$ ) it vertically; for $a < 0$ it is also reflected in the $x$ -axis. The point $Q(1 \mid a)$ always lies on the graph because $1^n = 1$ .

Depending on $n$ there are four classes (shown in different colours in the simulation):

Exponent	Graph (for $a = 1$ )	Symmetry	Domain $D$	Range $W$
$n$ even, $n > 0$	parabola of order $n$ , both branches up	about the $y$ -axis	$\mathbb{R}$	$\mathbb{R}_0^+$
$n$ odd, $n > 0$	parabola of order $n$ (for $n = 1$ a line through the origin), from bottom left to top right	about the origin	$\mathbb{R}$	$\mathbb{R}$
$n$ even, $n < 0$	hyperbola, both branches up	about the $y$ -axis	$\mathbb{R} \setminus \{0\}$	$\mathbb{R}^+$
$n$ odd, $n < 0$	hyperbola, branches in quadrants I and III	about the origin	$\mathbb{R} \setminus \{0\}$	$\mathbb{R} \setminus \{0\}$

Symmetry: For even $n$ , $f(-x) = f(x)$ : the graph is symmetric about the $y$ -axis. For odd $n$ , $f(-x) = -f(x)$ : the graph is symmetric about the origin.
Negative exponents: $x^{-n} = \frac{1}{x^n}$ . The function is not defined at $x = 0$ (pole); the axes $x = 0$ and $y = 0$ are asymptotes.
Special cases: $n = 1$ gives the line $y = a \cdot x$ through the origin. For $n = 0$ , $x^0 = 1$ (for $x \neq 0$ ), so $f(x) = a$ : a line parallel to the $x$ -axis. The expression $0^0$ is not considered here – the simulation leaves $0$ out of the domain (hollow point).
Common points: All graphs of $y = x^n$ pass through $(1 \mid 1)$ . For even $n$ they pass through $(-1 \mid 1)$ , for odd $n$ through $(-1 \mid -1)$ , and for $n > 0$ through the origin.
Between $-1$ and $1$ : the larger the (positive) exponent, the flatter the graph, e.g. $0.5^4 = 0.0625 < 0.5^2 = 0.25$ . Outside this interval it is the other way round.

Try it

Move the slider for $n$ slowly from $-4$ to $6$ , or click Run through all exponents. The graph morphs smoothly. For which $n$ does the class change? Watch the colour and the label at the top right of the key features.
Click Show symmetry for $n = 3$ : the right part is rotated through $180°$ about the origin. Try the same for $n = 4$ . Load example
Drag the point $Q(1 \mid a)$ up and down. What happens to the range and the monotonicity for $a < 0$ ? Load example
Drag $P$ along the graph and compare the calculation in the picture with the table of values. At the positions in the table, the matching row is highlighted.
Choose the family view and tap the individual exponents. The other graphs of the family are faint and labelled with their exponent. Which points always stay the same? Even exponents · Odd exponents
Compare $x$ and $x^3$ and let the vertical line run with Compare from left to right. Where does $x^3$ lie above $x$ ? Load example
Set $n = -1$ and $n = -2$ . How do the function values behave near $0$ on the left and on the right? Load example

Tasks

Task 1: Checking points

Do the points $A(-2 \mid -4)$ and $B(3 \mid 13)$ lie on the graph of $f(x) = 0.5 \cdot x^3$ ?

Show solution

$f(-2) = 0.5 \cdot (-2)^3 = 0.5 \cdot (-8) = -4$ , so $A$ lies on the graph. $f(3) = 0.5 \cdot 27 = 13.5 \neq 13$ , so $B$ does not. Show in the simulation

Task 2: Finding the stretch factor

The graph of $f(x) = a \cdot x^4$ passes through the point $(2 \mid 8)$ . Find $a$ and state the range. Load task

Show solution

$a \cdot 2^4 = 8 \Rightarrow 16a = 8 \Rightarrow a = 0.5$ . Because $n$ is even and $a > 0$ , the range is $W = \mathbb{R}_0^+$ . Show in the simulation

Task 3: Key features

Describe $f(x) = -2 \cdot x^{-3}$ : domain, range, symmetry, monotonicity, behaviour as $x \to \pm\infty$ and near $0$ . Compute $f(1)$ and $f(-1)$ .

Show solution

$D = W = \mathbb{R} \setminus \{0\}$ , symmetric about the origin ( $n$ odd). Because $a < 0$ , the hyperbola $\frac{1}{x^3}$ is reflected in the $x$ -axis: the branches lie in quadrants II and IV, and $f$ is increasing for $x < 0$ and for $x > 0$ . As $x \to \pm\infty$ , $f(x) \to 0$ ; to the left of $0$ , $f(x) \to +\infty$ , to the right $f(x) \to -\infty$ . Asymptotes $x = 0$ and $y = 0$ . $f(1) = -2$ , $f(-1) = -2 \cdot \frac{1}{-1} = 2$ . Show in the simulation

Task 4: Which is bigger?

For which $x$ is $x^4 < x^2$ ? Give a numerical example.

Show solution

The graphs intersect at $x = -1$ , $0$ and $1$ . For $0 < |x| < 1$ , $x^4 < x^2$ , e.g. $0.5^4 = 0.0625 < 0.25 = 0.5^2$ . For $|x| > 1$ it is the other way round, e.g. $2^4 = 16 > 4$ . Show in the simulation

Task 5: Common points

Which points do all graphs of $y = x^n$ with odd $n > 0$ have in common? Which one is lost if $n = -1$ and $n = -3$ are also allowed?

Show solution

$(-1 \mid -1)$ , $(0 \mid 0)$ and $(1 \mid 1)$ , because $(-1)^n = -1$ , $0^n = 0$ and $1^n = 1$ . For negative $n$ , $0$ is not in the domain, so the origin is lost. Show in the simulation

Task 6: Recognising a graph

A graph of $f(x) = x^n$ lies entirely above the $x$ -axis, is symmetric about the $y$ -axis and has a pole at $x = 0$ . What do you know about $n$ ? Give an example.

Show solution

$n$ is even and negative, e.g. $n = -2$ : $f(x) = \frac{1}{x^2}$ with $W = \mathbb{R}^+$ . Show in the simulation

Ideas for the classroom

Use: About grades 9/10, power functions with natural exponents and extending powers, then polynomial functions. Negative exponents build on the hyperbola $y = \frac{1}{x}$ from Year 8; please check the curriculum for whether and how far they are covered.
Discovery: The family view is a good way to collect conjectures (“What do they all have in common?”), which can then be justified with $1^n$ , $(-1)^n$ and $0^n$ .
Justifying symmetry: The animation shows what symmetry about an axis and about a point mean; the reasoning $(-x)^n = x^n$ or $(-x)^n = -x^n$ can follow.
Common stumbling blocks:
- “ $x^4$ is always bigger than $x^2$ .” – Only for $|x| > 1$ ; between $-1$ and $1$ it is the other way round.
- $x^{-2}$ is read as $-x^2$ . – In fact $x^{-2} = \frac{1}{x^2} > 0$ .
- For $n = 0$ , $0^0$ is simply set to $1$ . – The simulation deliberately leaves out $0$ ; a short discussion is worthwhile here.
- Monotonicity for odd negative $n$ : $\frac{1}{x}$ is decreasing on each part, but not on the whole domain (e.g. $f(-1) = -1 < f(1) = 1$ ).
Task mode: With info=0 and Share → Hide results, the key features, the table of values and the calculation are replaced by “?”; the graph stays visible.