# Power Functions

### Graphs of Power Functions A power function has the form $f(x) = ax^n$, where $a$ is a nonzero real number and $n$ is a real number.
A power function is a function, where $a$ is a nonzero real number and $n$ is a real number, written in the form:
$f(x) = ax^n$
The degree of the power function is the value of $n$, the exponent on the variable. The coefficient $a$ and the degree $n$ determine the shape of the graph. For power functions where $n$ is a positive integer, the domain is all real numbers. The linear function $f(x)=x$, the quadratic function $f(x)=x^2$, and the cubic function $f(x)=x^3$ are some examples of power functions.

For power functions with positive integer powers:

• A power function of the form $f(x)=x^n$ is a parent function. The value of $n$ determines the shape of the graph.
• The graph of each parent function with a positive integer power contains the points $(0,0)$ and $(1,1)$. As nnn increases, the graph of a power function f(x)=xnf(x)=x^nf(x)=xn becomes flatter for values of xxx between -1 and 1, and steeper for values of xxx greater than 1 and less than -1.

### Transformations of Power Functions The graph of a power function can be transformed by performing operations on the function rule.

Transformations can be used to make changes to the graph of a parent function. These include translations (shifts), stretches or compressions, and reflections.

For any function $f(x)$, the function can be translated vertically $k$ units and translated horizontally $h$ units. Horizontal translations are opposite in direction from the sign: Subtracting $h$ from the input translates the graph in the positive direction, and adding $h$ to the input translates it in the negative direction.

A function $f(x)$ can also be stretched or compressed by a factor of $a$ and reflected across the $x$-axis.

Vertical Translations Horizontal Translations
For the graph of $f(x)=x^n$ and $k>0$:
• The graph of $f(x)+k=x^n+k$ is translated up by $k$ units.
• The graph of $f(x)-k=x^n-k$ is translated down by $k$ units.
For the graph of $f(x)=x^n$ and $h>0$:
• The graph of $f(x-h)=(x-h)^n$ is translated right by $h$ units.
• The graph of $f(x+h)=(x+h)^n$ is translated left by $h$ units.

Stretches and Compressions Reflections
For the graph of $f(x)=x^n$ and $a>0$:
• The graph of $af(x)=ax^n$ is a vertical stretch of the graph of $f(x)$ by a factor of $a$ if $a>1$.
• The graph of $af(x)=ax^n$ is a vertical compression of the graph of $f(x)$ by a factor of $a$ if $0.
For the graph of $f(x)$:
• The graph of $-f(x)=-x^n$ is a reflection of the graph of $f(x)$ across the $x$-axis.

Step-By-Step Example
Graphing Transformations of a Polynomial Function
Graph the given function:
$f(x)=-(x+3)^3-1$
Step 1

Identify the parent function.

The first term in the polynomial is cubed.
$-(x+3)^{3}$
So it has the form of a cubic polynomial. The parent function of a cubic polynomial is:
$f(x)=x^3$
Step 2
Sketch the graph of the parent function. The parent functions of all power functions contain the points $(0,0)$ and $(1,1)$. The shape of a cubic function curves up in the first quadrant, somewhat like a parabola, and down in the third quadrant.
Step 3
Identify any stretches, compressions, or reflections. The given function is in the form:
\begin{aligned}f(x)&=-a(x+h)^3-k\\f(x)&=-(x+3)^3-1\end{aligned}
The negative sign indicates that the parent function is reflected across the $x$-axis.

If the parent function is multiplied by $a$, where $a>1$, then the function is stretched. If $0, then the function is compressed. The value of $a=1$ in the given function indicates that the parent function is not stretched or compressed.

Step 4
Sketch the reflection of the parent function.
Step 5
Identify any translations. The given function shows a translation to the left by $h$, or 3 units:
\begin{aligned}f(x+h)&=(x+{\color{#c42126}{h}})^3\\f(x)&=-(x+{\color{#c42126}{3}})^3-1\end{aligned}
The given function also shows a translation down by $k$, or 1 unit:
\begin{aligned}f(x)-k&=x^3-{\color{#c42126}{k}}\\f(x)&=-(x+3)^3-{\color{#c42126}{1}} \end{aligned}
Solution
Since $h=3$ and $k=1$, the graph of the reflected function is translated left by 3 units and down by 1 unit. Sketch the graph of the given function.