The power functions

The power functions are functions of the form

\boxed{f(x)=A(x-v)^p+B}

where $A, v$ and $b$ are constants, and the exponent $p$ can be an arbitrary number. Note that for $p=2$ we obtain the vertex form of the quadratic function. Thus, power function are a generalisation of the quadratic functions.

Let's quickly discuss the trivial cases:

for $p=0$ we have

f(x)=A(x-v)^0+B = A\cdot 1+B=A+B

which is a constant function,

for $p=1$ we get the linear function

f(x)=A(x-v)^1+B = Ax+(Av+B)

which is of the form $f(x)=ax+b$ .

For further discussion, we therefore assume that $p$ is different form $0$ and $1$ . In particular, we will discuss the cases where $p$ is either a natural number, a fraction, or a negative integer: ( $p=2,3,4,...$ ) or a fraction ( $p=\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...$ ) or a negative number ( $p=...,-4,-3,-2,-1$ ):

Power functions with exponent $p=n$ , where $n\in \{2,3,4...\}$ :
$\boxed{f(x)=A(x-v)^n+B}$
e.g.
$\begin{array}{llll} f(x) & = & 3(x-2)^2-10 & (n=2)\\ f(x) & = & -(x-2)^{10}+1 & (n=10)\\ \end{array}$
Power functions with exponent $p=-n$ , where $n\in \{1, 2,3,4...\}$ :
$f(x)=A(x-v)^{-n}+B = \frac{A}{(x-v)^n}+B$
e.g.
$\begin{array}{llll} f(x) & = & \frac{3}{(x-2)^2}-10 & (n=2, p=-2)\\ f(x) & = & -\frac{1}{(x-2)^{10}}+1 & (n=10, p=-10)\\ \end{array}$
Power functions with exponent $p=\frac{1}{n}$ , where $n\in \{2,3,4...\}$ :
$f(x)=A(x-v)^{1/n}+B = A \sqrt[n]{x-v} +B$
e.g.
$\begin{array}{llll} f(x) & = & 3\cdot \sqrt{x-2}-10 & (n=2, p=1/2)\\ f(x) & = & -\sqrt[10]{x-2}+1 & (n=10, p=1/10)\\ \end{array}$

Exercise 1

Are these power functions? If so, determine $p$ , $A$ , $v$ and $B$ :

$f(x)=\frac{1}{x-2}$
$f(x)=7\cdot \sqrt[3]{x+5}-9$
$f(x)=-\frac{10}{(x+5.2)^3}+7$
$f(x)=2^{x-3}$

Solution

$f(x)=1\cdot (x-2)^{-1}+0$ , also $p=-1, A=1, v=2, B=0$
$f(x)=7\cdot (x+5)^{1/3}-9=7\cdot (x-(-5))^{1/3}-9$ , also $p=1/3, A=7, v=-5, B=-9$
$f(x)=-10\cdot (x+5.2)^{-3}+7=-10\cdot (x-(-5.2))^{-3}+7$ , also $p=-3, A=-10, v=-5.2, B=7$
This is an exponential function, as the input $x$ is in the exponent.

What do the graphs of power functions look like? We will first restrict ourselves to the reference functions (therefore A=1, v=0, B=0). Similar to the quadratic functions, the parameters $A$ , $B$ and $v$ have a geometrical interpretation:

$A$ stretches the graph of the reference function along the $y$ -direction,
$v$ moves the graph along the $x$ -direction, and
$B$ moves the graph along the $y$ -direction.

So we have three types of reference functions whose graphs we will now examine in more detail:

$r(x)=x^n$ , where $n=2,3,4,...$ The graph of these functions is called parabola.
$r(x)=x^{-n}=\frac{1}{x^n}$ where $n=1,2,3,4,...$ The graph of these functions is called hyperbola.
$r(x)=x^{1/n}=\sqrt[n]{x}$ where $n=2,3,4,...$

Exercise 2

Q1

Draw for each exercise the graphs of the reference functions $r_1$ and $r_2$ into the same coordinate system using a table of values. Important: Take your time and really use a table of values, don't take shortcuts! You will learn a lot from it.

$r_1(x)=x^2$ and $r_2(x)=x^3$
$r_1(x)=\frac{1}{x^2}$ and $r_2(x)=\frac{1}{x^3}$ What happens at $x=0$ ?
$r_1(x)=\sqrt{x}$ and $r_2(x)=\sqrt[3]{x}$

Q2

Which graphs in Q1 have no $x$ -intercept, which have no $y$ -intercept? Can this be generalised to higher values of $n$ ?

Q3

Notice that some graphs in Q1 have a negative branch and others do not. Why? Can this be generalised to higher values of $n$ ?

Q4

Which graphs in Q1 pass through the point

$O(0\vert 0)$
$P(1\vert 1)$
$Q(-1\vert -1)$
$R(-1\vert 1)$
$T(1\vert -1)$

Again, generalise for higher numbers of $n$ .

Solution

Let $r(x)=x^n$ or $r(x)=x^{-n}$ or $r(x)=x^{1/n}$ , where $n=1,2,3,...$ .

A1

Check your solutions using the Geogebra apps below.

A2

$r(x)=x^{1,2,3,...}$ : the graph passes through the origin and thus has an $x$ -intercept and a $y$ -intercept.
$r(x)=x^{...,-3,-2,-1}$ : $r$ has neither an $x$ - nor a $y$ -axis intercept. At $x=0$ there is no $y$ -value (it would be $\infty$ or $-\infty$ ).
$r(x)=x^{1/2, 1/3, 1/4,...}$ : the graph passes through the zero point and thus has an $x$ - and $y$ -intercept.

A3

The graph of $r$ has a negative branch only if $n$ is odd. This results from the fact that $(-1)^{odd}=-1$ and $(-1)^{even}=1$ .

A4

$O(0\vert 0)$ : all but the hyperbolas.
$P(1\vert 1)$ : all graphs
$Q(-1\vert -1)$ : all power functions $n=odd$ , $n=-odd$ , $n=1/odd$
$R(-1\vert 1)$ : hyperbolas and parabolas with $n$ even.
$T(1\vert -1)$ : none.

Exercise 3

The following should now be solvable: For each of the cases given below, make a rough sketch of the graph in a coordinate system. Use for each case a different coordinate system.

$r(x)=x^{\text{even}}$
$r(x)=x^{\text{odd}}$
$r(x)=x^{-\text{even}}=\frac{1}{x^\text{even}}$
$r(x)=x^{-\text{odd}}=\frac{1}{x^\text{odd}}$
$r(x)=x^{1/\text{even}}=\sqrt[\text{even}]{x}$
$r(x)=x^{1/\text{odd}}=\sqrt[\text{odd}]{x}$

Solution