Polynomial Functions and Modeling

Power Functions

Graphs of Power Functions

A power function has the form f(x)=axnf(x) = ax^n, where aa is a nonzero real number and nn is a real number.
A power function is a function, where aa is a nonzero real number and nn is a real number, written in the form:
f(x)=axnf(x) = ax^n
The degree of the power function is the value of nn, the exponent on the variable. The coefficient aa and the degree nn determine the shape of the graph. For power functions where nn is a positive integer, the domain is all real numbers. The linear function f(x)=xf(x)=x, the quadratic function f(x)=x2f(x)=x^2, and the cubic function f(x)=x3f(x)=x^3 are some examples of power functions.

For power functions with positive integer powers:

  • A power function of the form f(x)=xnf(x)=x^n is a parent function. The value of nn determines the shape of the graph.
  • The graph of each parent function with a positive integer power contains the points (0,0)(0,0) and (1,1)(1,1).
As nn increases, the graph of a power function f(x)=xnf(x)=x^n becomes flatter for values of xx between -1 and 1, and steeper for values of xx 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)f(x), the function can be translated vertically kk units and translated horizontally hh units. Horizontal translations are opposite in direction from the sign: Subtracting hh from the input translates the graph in the positive direction, and adding hh to the input translates it in the negative direction.

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

Vertical Translations Horizontal Translations
For the graph of f(x)=xnf(x)=x^n and k>0k>0:
  • The graph of f(x)+k=xn+kf(x)+k=x^n+k is translated up by kk units.
  • The graph of f(x)k=xnkf(x)-k=x^n-k is translated down by kk units.
For the graph of f(x)=xnf(x)=x^n and h>0h>0:
  • The graph of f(xh)=(xh)nf(x-h)=(x-h)^n is translated right by hh units.
  • The graph of f(x+h)=(x+h)nf(x+h)=(x+h)^n is translated left by hh units.

Stretches and Compressions Reflections
For the graph of f(x)=xnf(x)=x^n and a>0a>0:
  • The graph of af(x)=axnaf(x)=ax^n is a vertical stretch of the graph of f(x)f(x) by a factor of aa if a>1a>1.
  • The graph of af(x)=axnaf(x)=ax^n is a vertical compression of the graph of f(x)f(x) by a factor of aa if 0<a<10<a<1.
For the graph of f(x)f(x):
  • The graph of f(x)=xn-f(x)=-x^n is a reflection of the graph of f(x)f(x) across the xx-axis.

Step-By-Step Example
Graphing Transformations of a Polynomial Function
Graph the given function:
f(x)=(x+3)31f(x)=-(x+3)^3-1
Step 1

Identify the parent function.

The first term in the polynomial is cubed.
(x+3)3-(x+3)^{3}
So it has the form of a cubic polynomial. The parent function of a cubic polynomial is:
f(x)=x3f(x)=x^3
Step 2
Sketch the graph of the parent function. The parent functions of all power functions contain the points (0,0)(0,0) and (1,1)(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:
f(x)=a(x+h)3kf(x)=(x+3)31\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 xx-axis.

If the parent function is multiplied by aa, where a>1a>1, then the function is stretched. If 0<a<10<a<1, then the function is compressed. The value of a=1a=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 hh, or 3 units:
f(x+h)=(x+h)3f(x)=(x+3)31\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 kk, or 1 unit:
f(x)k=x3kf(x)=(x+3)31\begin{aligned}f(x)-k&=x^3-{\color{#c42126}{k}}\\f(x)&=-(x+3)^3-{\color{#c42126}{1}} \end{aligned}
Solution
Since h=3h=3 and k=1k=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.