power function

A power function is a function in the form $f(x)=a[b(x-h){]}^n+k$, given that $n$ is rational. This means that $n$ can be expressed as an integer over an integer.

$l=a{b}^n(i-h{)}^n+k$

Best way to identify a power function is the linear inner expression.

Note: a power function is not an exponential function.

Figure: You must remember the 9 simple power functions of the form $f(x)=x^n$.

A power function is a function in the form $f(x)=a[b(x-h){]}^n+k$

• $a$ will make the graph reflect vertically if sign change.
• $b$ will make the graph reflect horizontally if sign change.
• $h$ is horizontal shift.
• $k$ is vertical shift from the origin.

for example, $f(x)=(2(x-1){)}^{-3}$ can be expressed as $f(x)=\frac{1}{[2(x-1){]}^3}$

when the exponent of a power function is a fraction we can use algebra to express the function differently
for example, $f(x)=(2(x-1){)}^{\frac{1}{3}}$ can be expressed as $f(x)=\sqrt[3]{2(x-1)}$

picture $f(a)=a^{\frac{1}{n}}=\frac{1}{a^n}$ where $a$ and $n$ are constants
if $n$ is even, $a^{\frac{1}{n}}$ must be positive

if $n$ is odd, $a^{\frac{1}{n}}$ must have the same sign as $a$

https://www.youtube.com/watch?v=Kr_Z61NVW-w
https://www.youtube.com/watch?v=GUzxZUzqJJ8